researchseminars.org

Narad Rampersad (University of Winnipeg)
2020-05-05 12:30 - 13:30 UTC | https://researchseminars.org/talk/OWNS/1/
Title: Ostrowski numeration and repetitions in words (https://researchseminars.org/talk/OWNS/1/)
by Narad Rampersad (University of Winnipeg) as part of One World Numeration seminar

Abstract
One of the classical results in combinatorics on words is Dejean's Theorem, which specifies the smallest exponent of repetitions that are avoidable on a given alphabet. One can ask if it is possible to determine this quantity (called the *repetition threshold*) for certain families of infinite words. For example, it is known that the repetition threshold for Sturmian words is 2+phi, and this value is reached by the Fibonacci word. Recently, this problem has been studied for *balanced words* (which generalize Sturmian words) and *rich words*. The infinite words constructed to resolve this problem can be defined in terms of the Ostrowski-numeration system for certain continued-fraction expansions. They can be viewed as *Ostrowski-automatic* sequences, where we generalize the notion of *k-automatic sequence* from the base-k numeration system to the Ostrowski numeration system.

Boris Solomyak (University of Bar-Ilan)
2020-05-19 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/2/
Title: On singular substitution Z-actions (https://researchseminars.org/talk/OWNS/2/)
by Boris Solomyak (University of Bar-Ilan) as part of One World Numeration seminar

Abstract
We consider primitive aperiodic substitutions on $d$ letters and the spectral properties of associated dynamical systems. In an earlier work we introduced a spectral cocycle, related to a kind of matrix Riesz product, which extends the (transpose) substitution matrix to the $d$-dimensional torus. The asymptotic properties of this cocycle provide local information on the (fractal) dimension of spectral measures. In the talk I will discuss a sufficient condition for the singularity of the spectrum in terms of the top Lyapunov exponent of this cocycle.

This is a joint work with A. Bufetov.

Olivier Carton (Université de Paris)
2020-05-12 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/3/
Title: Preservation of normality by selection (https://researchseminars.org/talk/OWNS/3/)
by Olivier Carton (Université de Paris) as part of One World Numeration seminar

Abstract
We first recall Agafonov's theorem which states that finite state selection preserves normality. We also give two slight extensions of this result to non-oblivious selection and suffix selection. We also propose a similar statement in the more general setting of shifts of finite type by defining selections which are compatible with the shift.

Célia Cisternino (University of Liège)
2020-05-26 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/4/
Title: Ergodic behavior of transformations associated with alternate base expansions (https://researchseminars.org/talk/OWNS/4/)
by Célia Cisternino (University of Liège) as part of One World Numeration seminar

Abstract
We consider a p-tuple of real numbers greater than 1, $\boldsymbol{\beta} = (\beta_1,\dots,\beta_p)$, called an alternate base, to represent real numbers. Since these representations generalize the 𝛽-representation introduced by Rényi in 1958, a lot of questions arise. In this talk, we will study the transformation generating the alternate base expansions (greedy representations). First, we will compare the $\boldsymbol{\beta}$-expansion and the $(\beta_1*\cdots*\beta_p)$-expansion over a particular digit set and study the cases when the equality holds. Next, we will talk about the existence of a measure equivalent to Lebesgue, invariant for the transformation corresponding to the alternate base and also about the ergodicity of this transformation.

This is a joint work with Émilie Charlier and Karma Dajani.

Simon Baker (University of Birmingham)
2020-06-09 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/6/
Title: Equidistribution results for self-similar measures (https://researchseminars.org/talk/OWNS/6/)
by Simon Baker (University of Birmingham) as part of One World Numeration seminar

Abstract
A well known theorem due to Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)$ is uniformly distributed modulo one. In this talk I will discuss an analogue of this statement that holds for fractal measures. As a corollary of this result we show that if $C$ is equal to the middle third Cantor set and $t\geq 1$, then almost every $x\in C+t$ is such that $(x^n)$ is uniformly distributed modulo one. Here almost every is with respect to the natural measure on $C+t$.

Carlos Matheus (Ecole Polytechnique)
2020-06-16 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/7/
Title: Approximations of the Lagrange and Markov spectra (https://researchseminars.org/talk/OWNS/7/)
by Carlos Matheus (Ecole Polytechnique) as part of One World Numeration seminar

Abstract
The Lagrange and Markov spectra are closed subsets of the positive real numbers defined in terms of diophantine approximations. Their topological structures are quite involved: they begin with an explicit discrete subset accumulating at $3$, they end with a half-infinite ray of the form $[4.52\cdots,\infty)$, and the portions between $3$ and $4.52\cdots$ contain complicated Cantor sets. In this talk, we describe polynomial time algorithms to approximate (in Hausdorff topology) these spectra.

Niels Langeveld (Leiden University)
2020-06-30 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/9/
Title: Continued fractions with two non integer digits (https://researchseminars.org/talk/OWNS/9/)
by Niels Langeveld (Leiden University) as part of One World Numeration seminar

Abstract
In this talk, we will look at a family of continued fraction expansions for which the digits in the expansions can attain two different (typically non-integer) values, named $\alpha_1$ and $\alpha_2$ with $\alpha_1 \alpha_2 \le 1/2$. If $\alpha_1 \alpha_2 < 1/2$ we can associate a dynamical system to these expansions with a switch region and therefore with lazy and greedy expansions. We will explore the parameter space and highlight certain values for which we can construct the natural extension (such as a family for which the lowest digit cannot be followed by itself). We end the talk with a list of open problems.

Hajime Kaneko (University of Tsukuba)
2020-07-07 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/10/
Title: Analogy of Lagrange spectrum related to geometric progressions (https://researchseminars.org/talk/OWNS/10/)
by Hajime Kaneko (University of Tsukuba) as part of One World Numeration seminar

Abstract
Classical Lagrange spectrum is defined by Diophantine approximation properties of arithmetic progressions. The theory of Lagrange spectrum is related to number theory and symbolic dynamics. In our talk we introduce significantly analogous results of Lagrange spectrum in uniform distribution theory of geometric progressions. In particular, we discuss the geometric sequences whose common ratios are Pisot numbers. For studying the fractional parts of geometric sequences, we introduce certain numeration system.

This talk is based on a joint work with Shigeki Akiyama.

Attila Pethő (University of Debrecen)
2020-07-14 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/11/
Title: On diophantine properties of generalized number systems - finite and periodic representations (https://researchseminars.org/talk/OWNS/11/)
by Attila Pethő (University of Debrecen) as part of One World Numeration seminar

Abstract
In this talk we investigate elements with special patterns in their representations in number systems in algebraic number fields. We concentrate on periodicity and on the representation of rational integers. We prove under natural assumptions that there are only finitely many $S$-units whose representation is periodic with a fixed period. We prove that the same holds for the set of values of polynomials at rational integers.

Derong Kong (Chongqing University)
2020-06-23 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/12/
Title: Univoque bases of real numbers: local dimension, Devil's staircase and isolated points (https://researchseminars.org/talk/OWNS/12/)
by Derong Kong (Chongqing University) as part of One World Numeration seminar

Abstract
Given a positive integer $M$ and a real number $x$, let $U(x)$ be the set of all bases $q \in (1,M+1]$ such that $x$ has a unique $q$-expansion with respect to the alphabet $\{0,1,\dots,M\}$. We will investigate the local dimension of $U(x)$ and prove a 'variation principle' for unique non-integer base expansions. We will also determine the critical values and the topological structure of $U(x)$.

Bill Mance (Adam Mickiewicz University in Poznań)
2020-09-01 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/13/
Title: Hotspot Lemmas for Noncompact Spaces (https://researchseminars.org/talk/OWNS/13/)
by Bill Mance (Adam Mickiewicz University in Poznań) as part of One World Numeration seminar

Abstract
We will explore a correction of several previously claimed generalizations of the classical hotspot lemma. Specifically, there is a common mistake that has been repeated in proofs going back more than 50 years. Corrected versions of these theorems are increasingly important as there has been more work in recent years focused on studying various generalizations of the concept of a normal number to numeration systems with infinite digit sets (for example, various continued fraction expansions, the Lüroth series expansion and its generalizations, and so on). Also, highlighting this (elementary) mistake may be helpful for those looking to study these numeration systems further and wishing to avoid some common pitfalls.

Bing Li (South China University of Technology)
2020-09-08 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/14/
Title: Some fractal problems in beta-expansions (https://researchseminars.org/talk/OWNS/14/)
by Bing Li (South China University of Technology) as part of One World Numeration seminar

Abstract
For greedy beta-expansions, we study some fractal sets of real numbers whose orbits under beta-transformation share some common properties. For example, the partial sum of the greedy beta-expansion converges with the same order, the orbit is not dense, the orbit is always far from that of another point etc. The usual tool is to approximate the beta-transformation dynamical system by Markov subsystems. We also discuss the similar problems for intermediate beta-expansions.

Jeffrey Shallit (University of Waterloo)
2020-09-15 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/15/
Title: Lazy Ostrowski Numeration and Sturmian Words (https://researchseminars.org/talk/OWNS/15/)
by Jeffrey Shallit (University of Waterloo) as part of One World Numeration seminar

Abstract
In this talk I will discuss a new connection between the so-called "lazy Ostrowski" numeration system, and periods of the prefixes of Sturmian characteristic words. I will also give a relationship between periods and the so-called "initial critical exponent". This builds on work of Frid, Berthé-Holton-Zamboni, Epifanio-Frougny-Gabriele-Mignosi, and others, and is joint work with Narad Rampersad and Daniel Gabric.

Yotam Smilansky (Rutgers University)
2020-09-22 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/16/
Title: Multiscale Substitution Tilings (https://researchseminars.org/talk/OWNS/16/)
by Yotam Smilansky (Rutgers University) as part of One World Numeration seminar

Abstract
Multiscale substitution tilings are a new family of tilings of Euclidean space that are generated by multiscale substitution rules. Unlike the standard setup of substitution tilings, which is a basic object of study within the aperiodic order community and includes examples such as the Penrose and the pinwheel tilings, multiple distinct scaling constants are allowed, and the defining process of inflation and subdivision is a continuous one. Under a certain irrationality assumption on the scaling constants, this construction gives rise to a new class of tilings, tiling spaces and tiling dynamical system, which are intrinsically different from those that arise in the standard setup. In the talk I will describe these new objects and discuss various structural, geometrical, statistical and dynamical results. Based on joint work with Yaar Solomon.

Marta Maggioni (Leiden University)
2020-09-29 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/17/
Title: Random matching for random interval maps (https://researchseminars.org/talk/OWNS/17/)
by Marta Maggioni (Leiden University) as part of One World Numeration seminar

Abstract
In this talk we extend the notion of matching for deterministic transformations to random matching for random interval maps. For a large class of piecewise affine random systems of the interval, we prove that this property of random matching implies that any invariant density of a stationary measure is piecewise constant. We provide examples of random matching for a variety of families of random dynamical systems, that includes generalised beta-transformations, continued fraction maps and a family of random maps producing signed binary expansions. We finally apply the property of random matching and its consequences to this family to study minimal weight expansions.
Based on a joint work with Karma Dajani and Charlene Kalle.

Francesco Veneziano (University of Genova)
2020-10-06 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/18/
Title: Finiteness and periodicity of continued fractions over quadratic number fields (https://researchseminars.org/talk/OWNS/18/)
by Francesco Veneziano (University of Genova) as part of One World Numeration seminar

Abstract
We consider continued fractions with partial quotients in the ring of integers of a quadratic number field $K$; a particular example of these continued fractions is the $\beta$-continued fraction introduced by Bernat. We show that for any quadratic Perron number $\beta$, the $\beta$-continued fraction expansion of elements in $\mathbb{Q}(\beta)$ is either finite of eventually periodic. We also show that for certain four quadratic Perron numbers $\beta$, the $\beta$-continued fraction represents finitely all elements of the quadratic field $\mathbb{Q}(\beta)$, thus answering questions of Rosen and Bernat.
Based on a joint work with Zuzana Masáková and Tomáš Vávra.

Kan Jiang (Ningbo University)
2020-10-13 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/19/
Title: Representations of real numbers on fractal sets (https://researchseminars.org/talk/OWNS/19/)
by Kan Jiang (Ningbo University) as part of One World Numeration seminar

Abstract
There are many approaches which can represent real numbers. For instance, the $\beta$-expansions, the continued fraction and so forth. Representations of real numbers on fractal sets were pioneered by H. Steinhaus who proved in 1917 that $C+C=[0,2]$ and $C−C=[−1,1]$, where $C$ is the middle-third Cantor set. Equivalently, for any $x \in [0,2]$, there exist some $y,z \in C$ such that $x=y+z$. In this talk, I will introduce similar results in terms of some fractal sets.

Paul Surer (University of Natural Resources and Life Sciences, Vienna)
2020-10-20 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/20/
Title: Representations for complex numbers with integer digits (https://researchseminars.org/talk/OWNS/20/)
by Paul Surer (University of Natural Resources and Life Sciences, Vienna) as part of One World Numeration seminar

Abstract
In this talk we present the zeta-expansion as a complex version of the well-known beta-expansion. It allows us to expand complex numbers with respect to a complex base by using integer digits. Our concepts fits into the framework of the recently published rotational beta-expansions. But we also establish relations with piecewise affine maps of the torus and with shift radix systems.

Mélodie Andrieu (Aix-Marseille University)
2020-10-27 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/21/
Title: A Rauzy fractal unbounded in all directions of the plane (https://researchseminars.org/talk/OWNS/21/)
by Mélodie Andrieu (Aix-Marseille University) as part of One World Numeration seminar

Abstract
Until 2001 it was believed that, as for Sturmian words, the imbalance of Arnoux-Rauzy words was bounded - or at least finite. Cassaigne, Ferenczi and Zamboni disproved this conjecture by constructing an Arnoux-Rauzy word with infinite imbalance, i.e. a word whose broken line deviates regularly and further and further from its average direction. Today, we hardly know anything about the geometrical and topological properties of these unbalanced Rauzy fractals. The Oseledets theorem suggests that these fractals are contained in a strip of the plane: indeed, if the Lyapunov exponents of the matricial product associated with the word exist, one of these exponents at least is nonpositive since their sum equals zero. This talk aims at disproving this belief.

Tomáš Vávra (University of Waterloo)
2020-11-03 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/22/
Title: Distinct unit generated number fields and finiteness in number systems (https://researchseminars.org/talk/OWNS/22/)
by Tomáš Vávra (University of Waterloo) as part of One World Numeration seminar

Abstract
A distinct unit generated field is a number field K such that every algebraic integer of the field is a sum of distinct units. In 2015, Dombek, Masáková, and Ziegler studied totally complex quartic fields, leaving 8 cases unresolved. Because in this case there is only one fundamental unit $u$, their method involved the study of finiteness in positional number systems with base u and digits arising from the roots of unity in $K$.

First, we consider a more general problem of positional representations with base beta with an arbitrary digit alphabet $D$. We will show that it is decidable whether a given pair $(\beta, D)$ allows eventually periodic or finite representations of elements of $O_K$.

We are then able to prove the conjecture that the 8 remaining cases indeed are distinct unit generated.

Pieter Allaart (University of North Texas)
2020-11-10 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/23/
Title: On the smallest base in which a number has a unique expansion (https://researchseminars.org/talk/OWNS/23/)
by Pieter Allaart (University of North Texas) as part of One World Numeration seminar

Abstract
For $x>0$, let $U(x)$ denote the set of bases $q \in (1,2]$ such that $x$ has a unique expansion in base $q$ over the alphabet $\{0,1\}$, and let $f(x)=\inf U(x)$. I will explain that the function $f(x)$ has a very complicated structure: it is highly discontinuous and has infinitely many infinite level sets. I will describe an algorithm for numerically computing $f(x)$ that often gives the exact value in just a small finite number of steps. The Komornik-Loreti constant, which is $f(1)$, will play a central role in this talk. This is joint work with Derong Kong, and builds on previous work by Kong (Acta Math. Hungar. 150(1):194--208, 2016).

Jacques Sakarovitch (Irif, CNRS, and Télécom Paris)
2020-11-17 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/24/
Title: The carry propagation of the successor function (https://researchseminars.org/talk/OWNS/24/)
by Jacques Sakarovitch (Irif, CNRS, and Télécom Paris) as part of One World Numeration seminar

Abstract
Given any numeration system, the carry propagation at an integer $N$ is the number of digits that change between the representation of $N$ and $N+1$. The carry propagation of the numeration system as a whole is the average carry propagations at the first $N$ integers, as $N$ tends to infinity, if this limit exists.

In the case of the usual base $p$ numeration system, it can be shown that the limit indeed exists and is equal to $p/(p-1)$. We recover a similar value for those numeration systems we consider and for which the limit exists.

The problem is less the computation of the carry propagation than the proof of its existence. We address it for various kinds of numeration systems: abstract numeration systems, rational base numeration systems, greedy numeration systems and beta-numeration. This problem is tackled with three different types of techniques: combinatorial, algebraic, and ergodic, each of them being relevant for different kinds of numeration systems.

This work has been published in Advances in Applied Mathematics 120 (2020). In this talk, we shall focus on the algebraic and ergodic methods.

Joint work with V. Berthé (Irif), Ch. Frougny (Irif), and M. Rigo (Univ. Liège).

Michael Barnsley (Australian National University)
2020-12-01 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/25/
Title: Rigid fractal tilings (https://researchseminars.org/talk/OWNS/25/)
by Michael Barnsley (Australian National University) as part of One World Numeration seminar

Abstract
I will describe recent work, joint with Louisa Barnsley and Andrew Vince, concerning a symbolic approach to self-similar tilings. This approach uses graph-directed iterated function systems to analyze both classical tilings and also generalized tilings of what may be unbounded fractal subsets of $\mathbb{R}^n$. A notion of rigid tiling systems is defined. Our key theorem states that when the system is rigid, all the conjugacies of the tilings can be described explicitly. In the seminar I hope to prove this for the case of standard IFSs.

Tanja Isabelle Schindler (Scuola Normale Superiore di Pisa)
2020-12-08 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/26/
Title: Limit theorems on counting large continued fraction digits (https://researchseminars.org/talk/OWNS/26/)
by Tanja Isabelle Schindler (Scuola Normale Superiore di Pisa) as part of One World Numeration seminar

Abstract
We establish a central limit theorem for counting large continued fraction digits $(a_n)$, that is, we count occurrences $\{a_n>b_n\}$, where $(b_n)$ is a sequence of positive integers. Our result improves a similar result by Philipp, which additionally assumes that bn tends to infinity. Moreover, we also show this kind of central limit theorem for counting the number of occurrences entries such that the continued fraction entry lies between $d_n$ and $d_n(1+1/c_n)$ for given sequences $(c_n)$ and $(d_n)$. For such intervals we also give a refinement of the famous Borel–Bernstein theorem regarding the event that the nth continued fraction digit lying infinitely often in this interval. As a side result, we explicitly determine the first $\phi$-mixing coefficient for the Gauss system - a result we actually need to improve Philipp's theorem. This is joint work with Marc Kesseböhmer.

Lukas Spiegelhofer (Montanuniversität Leoben)
2020-12-15 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/27/
Title: The digits of $n+t$ (https://researchseminars.org/talk/OWNS/27/)
by Lukas Spiegelhofer (Montanuniversität Leoben) as part of One World Numeration seminar

Abstract
We study the binary sum-of-digits function $s_2$ under addition of a constant $t$.
For each integer $k$, we are interested in the asymptotic density $\delta(k,t)$ of integers $t$ such that $s_2(n+t)-s_2(n)=k$.
In this talk, we consider the following two questions.

(1) Do we have \[ c_t=\delta(0,t)+\delta(1,t)+\cdots>1/2? \]
This is a conjecture due to T. W. Cusick (2011).

(2) What does the probability distribution defined by $k\mapsto \delta(k,t)$ look like?

We prove that indeed $c_t>1/2$ if the binary expansion of $t$ contains at least $M$ blocks of contiguous ones, where $M$ is effective.
Our second theorem states that $\delta(j,t)$ usually behaves like a normal distribution, which extends a result by Emme and Hubert (2018).

This is joint work with Michael Wallner (TU Wien).

Claire Merriman (Ohio State University)
2021-01-05 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/28/
Title: $\alpha$-odd continued fractions (https://researchseminars.org/talk/OWNS/28/)
by Claire Merriman (Ohio State University) as part of One World Numeration seminar

Abstract
The standard continued fraction algorithm come from the Euclidean algorithm. We can also describe this algorithm using a dynamical system of $[0,1)$, where the transformation that takes $x$ to the fractional part of $1/x$ is said to generate the continued fraction expansion of $x$. From there, we ask two questions: What happens to the continued fraction expansion when we change the domain to something other than $[0,1)$? What happens to the dynamical system when we impose restrictions on the continued fraction expansion, such as finding the nearest odd integer instead of the floor? This talk will focus on the case where we first restrict to odd integers, then start shifting the domain $[\alpha-2, \alpha)$.

This talk is based on joint work with Florin Boca and animations done by Xavier Ding, Gustav Jennetten, and Joel Rozhon as part of an Illinois Geometry Lab project.

Tom Kempton (University of Manchester)
2021-01-19 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/29/
Title: Bernoulli Convolutions and Measures on the Spectra of Algebraic Integers (https://researchseminars.org/talk/OWNS/29/)
by Tom Kempton (University of Manchester) as part of One World Numeration seminar

Abstract
Given an algebraic integer $\beta$ and alphabet $A=\{-1,0,1\}$, the spectrum of $\beta$ is the set
$$\Sigma(\beta) :=\bigg\{\sum_{i=1}^n a_i\beta^i : n\in\mathbb N, a_i\in A\bigg\}.$$
In the case that $\beta$ is Pisot one can study the spectrum of $\beta$ dynamically using substitutions or cut and project schemes, and this allows one to see lots of local structure in the spectrum. There are higher dimensional analogues for other algebraic integers.

In this talk we will define a random walk on the spectrum of $\beta$ and show how, with appropriate renormalisation, this leads to an infinite stationary measure on the spectrum. This measure has local structure analagous to that of the spectrum itself. Furthermore, this measure has deep links with the Bernoulli convolution, and in particular new criteria for the absolute continuity of Bernoulli convolutions can be stated in terms of the ergodic properties of these measures.

Carlo Carminati (Università di Pisa)
2021-01-26 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/30/
Title: Prevalence of matching for families of continued fraction algorithms: old and new results (https://researchseminars.org/talk/OWNS/30/)
by Carlo Carminati (Università di Pisa) as part of One World Numeration seminar

Abstract
We will give an overview of the phenomenon of matching, which was first observed in the family of Nakada's $\alpha$-continued fractions, but is also encountered in other families of continued fraction algorithms.

Our main focus will be the matching property for the family of Ito-Tanaka continued fractions: we will discuss the analogies with Nakada's case
(such as prevalence of matching), but also some unexpected features which are peculiar of this case.

The core of the talk is about some recent results obtained in collaboration with Niels Langeveld and Wolfgang Steiner.

Samuel Petite (Université de Picardie Jules Verne)
2021-02-02 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/31/
Title: Interplay between finite topological rank minimal Cantor systems, $S$-adic subshifts and their complexity (https://researchseminars.org/talk/OWNS/31/)
by Samuel Petite (Université de Picardie Jules Verne) as part of One World Numeration seminar

Abstract
The family of minimal Cantor systems of finite topological rank includes Sturmian subshifts, coding of interval exchange transformations, odometers and substitutive subshifts. They are known to have dynamical rigidity properties. In a joint work with F. Durand, S. Donoso and A. Maass, we provide a combinatorial characterization of such subshifts in terms of S-adic systems. This enables to obtain some links with the factor complexity function and some new rigidity properties depending on the rank of the system.

Clemens Müllner (TU Wien)
2021-02-09 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/32/
Title: Multiplicative automatic sequences (https://researchseminars.org/talk/OWNS/32/)
by Clemens Müllner (TU Wien) as part of One World Numeration seminar

Abstract
It was shown by Mariusz Lemańczyk and the author that automatic sequences are orthogonal to bounded and aperiodic multiplicative functions. This is a manifestation of the disjointedness of additive and multiplicative structures. We continue this path by presenting in this talk a complete classification of complex-valued sequences which are both multiplicative and automatic. This shows that the intersection of these two worlds has a very special (and simple) form. This is joint work with Mariusz Lemańczyk and Jakub Konieczny.

Gerardo González Robert (Universidad Nacional Autónoma de México)
2021-02-16 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/33/
Title: Good's Theorem for Hurwitz Continued Fractions (https://researchseminars.org/talk/OWNS/33/)
by Gerardo González Robert (Universidad Nacional Autónoma de México) as part of One World Numeration seminar

Abstract
In 1887, Adolf Hurwitz introduced a simple procedure to write any complex number as a continued fraction with Gaussian integers as partial denominators and with partial numerators equal to 1. While similarities between regular and Hurwitz continued fractions abound, there are important differences too (for example, as shown in 1974 by R. Lakein, Serret's theorem on equivalent numbers does not hold in the complex case). In this talk, after giving a short overview of the theory of Hurwitz continued fractions, we will state and sketch the proof of a complex version of I. J. Good's theorem on the Hausdorff dimension of the set of real numbers whose regular continued fraction tends to infinity. Finally, we will discuss some open problems.

Seulbee Lee (Scuola Normale Superiore di Pisa)
2021-02-23 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/34/
Title: Odd-odd continued fraction algorithm (https://researchseminars.org/talk/OWNS/34/)
by Seulbee Lee (Scuola Normale Superiore di Pisa) as part of One World Numeration seminar

Abstract
The classical continued fraction gives the best approximating rational numbers of an irrational number. We define a new continued fraction, say odd-odd continued fraction, which gives the best approximating rational numbers whose numerators and denominators are odd. We see that a jump transformation associated to the Romik map induces the odd-odd continued fraction. We discuss properties of the odd-odd continued fraction expansions. This is joint work with Dong Han Kim and Lingmin Liao.

Vitaly Bergelson (Ohio State University)
2021-03-02 15:00 - 16:00 | https://researchseminars.org/talk/OWNS/35/
Title: Normal sets in $(\mathbb{ℕ},+)$ and $(\mathbb{N},\times)$ (https://researchseminars.org/talk/OWNS/35/)
by Vitaly Bergelson (Ohio State University) as part of One World Numeration seminar

Abstract
We will start with discussing the general idea of a normal set in a countable cancellative amenable semigroup, which was introduced and developed in the recent paper "A fresh look at the notion of normality" (joint work with Tomas Downarowicz and Michał Misiurewicz). We will move then to discussing and juxtaposing combinatorial and Diophantine properties of normal sets in semigroups $(\mathbb{ℕ},+)$ and $(\mathbb{N},\times)$. We will conclude the lecture with a brief review of some interesting open problems.

Natalie Priebe Frank (Vassar College)
2021-03-09 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/36/
Title: The flow view and infinite interval exchange transformation of a recognizable substitution (https://researchseminars.org/talk/OWNS/36/)
by Natalie Priebe Frank (Vassar College) as part of One World Numeration seminar

Abstract
A flow view is the graph of a measurable conjugacy between a substitution or S-adic subshift or tiling space and an exchange of infinitely many intervals in [0,1]. The natural refining sequence of partitions of the sequence space is transferred to [0,1] with Lebesgue measure using a canonical addressing scheme, a fixed dual substitution, and a shift-invariant probability measure. On the flow view, sequences are shown horizontally at a height given by their image under conjugacy.

In this talk I'll explain how it all works and state some results and questions. There will be pictures.

Alexandra Skripchenko (Higher School of Economics)
2021-03-16 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/37/
Title: Double rotations and their ergodic properties (https://researchseminars.org/talk/OWNS/37/)
by Alexandra Skripchenko (Higher School of Economics) as part of One World Numeration seminar

Abstract
Double rotations are the simplest subclass of interval translation mappings. A double rotation is of finite type if its attractor is an interval and of infinite type if it is a Cantor set. It is easy to see that the restriction of a double rotation of finite type to its attractor is simply a rotation. It is known due to Suzuki - Ito - Aihara and Bruin - Clark that double rotations of infinite type are defined by a subset of zero measure in the parameter set. We introduce a new renormalization procedure on double rotations, which is reminiscent of the classical Rauzy induction. Using this renormalization we prove that the set of parameters which induce infinite type double rotations has Hausdorff dimension strictly smaller than 3. Moreover, we construct a natural invariant measure supported on these parameters and show that, with respect to this measure, almost all double rotations are uniquely ergodic. In my talk I plan to outline this proof that is based on the recent result by Ch. Fougeron for simplicial systems. I also hope to discuss briefly some challenging open questions and further research plans related to double rotations.

The talk is based on a joint work with Mauro Artigiani, Charles Fougeron and Pascal Hubert.

Godofredo Iommi (Pontificia Universidad Católica de Chile)
2021-03-23 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/38/
Title: Arithmetic averages and normality in continued fractions (https://researchseminars.org/talk/OWNS/38/)
by Godofredo Iommi (Pontificia Universidad Católica de Chile) as part of One World Numeration seminar

Abstract
Every real number can be written as a continued fraction. There exists a dynamical system, the Gauss map, that acts as the shift in the expansion. In this talk, I will comment on the Hausdorff dimension of two types of sets: one of them defined in terms of arithmetic averages of the digits in the expansion and the other related to (continued fraction) normal numbers. In both cases, the non compactness that steams from the fact that we use countable many partial quotients in the continued fraction plays a fundamental role. Some of the results are joint work with Thomas Jordan and others together with Aníbal Velozo.

Michael Drmota (TU Wien)
2021-03-30 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/39/
Title: (Logarithmic) Densities for Automatic Sequences along Primes and Squares (https://researchseminars.org/talk/OWNS/39/)
by Michael Drmota (TU Wien) as part of One World Numeration seminar

Abstract
It is well known that the every letter $\alpha$ of an automatic sequence $a(n)$ has
a logarithmic density -- and it can be decided when this logarithmic density is actually a density.
For example, the letters $0$ and $1$ of the Thue-Morse sequences $t(n)$ have both frequences $1/2$.
[The Thue-Morse sequence is the binary sum-of-digits functions modulo 2.]

The purpose of this talk is to present a corresponding result for subsequences of general
automatic sequences along primes and squares. This is a far reaching generalization of two breakthrough
results of Mauduit and Rivat from 2009 and 2010, where they solved two conjectures by Gelfond
on the densities of $0$ and $1$ of $t(p_n)$ and $t(n^2)$ (where $p_n$ denotes the sequence of primes).

More technically, one has to develop a method to transfer density results for primitive automatic
sequences to logarithmic-density results for general automatic sequences. Then as an application
one can deduce that the logarithmic densities of any automatic sequence along squares
$(n^2)_{n\geq 0}$ and primes $(p_n)_{n\geq 1}$ exist and are computable.
Furthermore, if densities exist then they are (usually) rational.

Andrew Mitchell (University of Birmingham)
2021-04-13 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/40/
Title: Measure theoretic entropy of random substitutions (https://researchseminars.org/talk/OWNS/40/)
by Andrew Mitchell (University of Birmingham) as part of One World Numeration seminar

Abstract
Random substitutions and their associated subshifts provide a model for structures that exhibit both long range order and positive topological entropy. In this talk we discuss the entropy of a large class of ergodic measures, known as frequency measures, that arise naturally from random substitutions. We introduce a new measure of complexity, namely measure theoretic inflation word entropy, and discuss its relationship to measure theoretic entropy. This new measure of complexity provides a framework for the systematic study of measure theoretic entropy for random substitution subshifts.

As an application of our results, we obtain closed form formulas for the entropy of frequency measures for a wide range of random substitution subshifts and show that in many cases there exists a frequency measure of maximal entropy. Further, for a class of random substitution subshifts, we show that this measure is the unique measure of maximal entropy.

This talk is based on joint work with P. Gohlke, D. Rust, and T. Samuel.

Ayreena Bakhtawar (La Trobe University)
2021-04-20 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/41/
Title: Metrical theory for the set of points associated with the generalized Jarnik-Besicovitch set (https://researchseminars.org/talk/OWNS/41/)
by Ayreena Bakhtawar (La Trobe University) as part of One World Numeration seminar

Abstract
From Lagrange's (1770) and Legendre's (1808) results we conclude that to find good rational approximations to an irrational number we only need to focus on its convergents. Let $[a_1(x),a_2(x),\dots]$ be the continued fraction expansion of a real number $x \in [0,1)$. The Jarnik-Besicovitch set in terms of continued fraction consists of all those $x \in [0,1)$ which satisfy $a_{n+1}(x) \ge e^{\tau\, (\log|T'x|+⋯+\log|T'(T^{n-1}x)|)}$ for infinitely many $n \in \mathbb{N}$, where $a_{n+1}(x)$ is the $(n+1)$-th partial quotient of $x$ and $T$ is the Gauss map. In this talk, I will focus on determining the Hausdorff dimension of the set of real numbers $x \in [0,1)$ such that for any $m \in \mathbb{N}$ the following holds for infinitely many $n \in \mathbb{N}$: $a_{n+1}(x) a_{n+2}(x) \cdots a_{n+m}(x) \ge e^{τ(x)\, (f(x)+⋯+f(T^{n-1}x))}$, where $f$ and $\tau$ are positive continuous functions. Also we will see that for appropriate choices of $m$, $\tau(x)$ and $f(x)$ our result implies various classical results including the famous Jarnik-Besicovitch theorem.

Boris Adamczewski (CNRS, Université Claude Bernard Lyon 1)
2021-04-27 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/42/
Title: Expansions of numbers in multiplicatively independent bases: Furstenberg's conjecture and finite automata (https://researchseminars.org/talk/OWNS/42/)
by Boris Adamczewski (CNRS, Université Claude Bernard Lyon 1) as part of One World Numeration seminar

Abstract
It is commonly expected that expansions of numbers in multiplicatively independent bases, such as 2 and 10, should have no common structure. However, it seems extraordinarily difficult to confirm this naive heuristic principle in some way or another. In the late 1960s, Furstenberg suggested a series of conjectures, which became famous and aim to capture this heuristic. The work I will discuss in this talk is motivated by one of these conjectures. Despite recent remarkable progress by Shmerkin and Wu, it remains totally out of reach of the current methods. While Furstenberg’s conjectures take place in a dynamical setting, I will use instead the language of automata theory to formulate some related problems that formalize and express in a different way the same general heuristic. I will explain how the latter can be solved thanks to some recent advances in Mahler’s method; a method in transcendental number theory initiated by Mahler at the end of the 1920s. This a joint work with Colin Faverjon.

Tushar Das (University of Wisconsin - La Crosse)
2021-05-04 14:00 - 15:00 | https://researchseminars.org/talk/OWNS/43/
Title: Hausdorff Hensley Good & Gauss (https://researchseminars.org/talk/OWNS/43/)
by Tushar Das (University of Wisconsin - La Crosse) as part of One World Numeration seminar

Abstract
Several participants of the One World Numeration Seminar (OWNS) will know Hensley's haunting bounds (c. 1990) for the dimension of irrationals whose regular continued fraction expansion partial quotients are all at most N; while some might remember Good's great bounds (c. 1940) for the dimension of irrationals whose partial quotients are all at least N. We will report on relatively recent results in https://arxiv.org/abs/2007.10554 that allow one to extend such fabulous formulae to unexpected expansions. Our technology may be utilized to study various systems arising from numeration, dynamics, or geometry. The talk will be accessible to students and beyond, and I hope to present a sampling of open questions and research directions that await exploration.

Giulio Tiozzo (University of Toronto)
2021-05-11 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/44/
Title: The bifurcation locus for numbers of bounded type (https://researchseminars.org/talk/OWNS/44/)
by Giulio Tiozzo (University of Toronto) as part of One World Numeration seminar

Abstract
We define a family $B(t)$ of compact subsets of the unit interval which provides a filtration of the set of numbers whose continued fraction expansion has bounded digits. This generalizes to a continuous family the well-known sets of numbers whose continued fraction expansion is bounded above by a fixed integer.

We study how the set $B(t)$ changes as the parameter $t$ ranges in $[0,1]$, and describe precisely the bifurcations that occur as the parameters change. Further, we discuss continuity properties of the Hausdorff dimension of $B(t)$ and its regularity.

Finally, we establish a precise correspondence between these bifurcations and
the bifurcations for the classical family of real quadratic polynomials.

Joint with C. Carminati.

Joseph Vandehey (University of Texas at Tyler)
2021-05-18 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/45/
Title: Solved and unsolved problems in normal numbers (https://researchseminars.org/talk/OWNS/45/)
by Joseph Vandehey (University of Texas at Tyler) as part of One World Numeration seminar

Abstract
We will survey a variety of problems on normal numbers, some old, some new, some solved, and some unsolved, in the hope of spurring some new directions of study. Topics will include constructions of normal numbers, normality in two different systems simultaneously, normality seen through the lens of informational or logical complexity, and more.

Charles Fougeron (Université de Paris)
2021-05-25 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/46/
Title: Dynamics of simplicial systems and multidimensional continued fraction algorithms (https://researchseminars.org/talk/OWNS/46/)
by Charles Fougeron (Université de Paris) as part of One World Numeration seminar

Abstract
Motivated by the richness of the Gauss algorithm which allows to efficiently compute the best approximations of a real number by rationals, many mathematicians have suggested generalisations to study Diophantine approximations of vectors in higher dimensions. Examples include Poincaré's algorithm introduced at the end of the 19th century or those of Brun and Selmer in the middle of the 20th century. Since the beginning of the 90's to the present day, there has been many works studying the convergence and dynamics of these multidimensional continued fraction algorithms. In particular, Schweiger and Broise have shown that the approximation sequence built using Selmer and Brun algorithms converge to the right vector with an extra ergodic property. On the other hand, Nogueira demonstrated that the algorithm proposed by Poincaré almost never converges.

Starting from the classical case of Farey's algorithm, which is an "additive" version of Gauss's algorithm, I will present a combinatorial point of view on these algorithms which allows to us to use a random walk approach. In this model, taking a random vector for the Lebesgue measure will correspond to following a random walk with memory in a labelled graph called symplicial system. The laws of probability for this random walk are elementary and we can thus develop probabilistic techniques to study their generic dynamical behaviour. This will lead us to describe a purely graph theoretic criterion to check the convergence of a continued fraction algorithm.

Bastián Espinoza (Université de Picardie Jules Verne and Universidad de Chile)
2021-06-01 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/47/
Title: Automorphisms and factors of finite topological rank systems (https://researchseminars.org/talk/OWNS/47/)
by Bastián Espinoza (Université de Picardie Jules Verne and Universidad de Chile) as part of One World Numeration seminar

Abstract
Finite topological rank systems are a type of minimal S-adic subshift that includes many of the classical minimal systems of zero entropy (e.g. linearly recurrent subshifts, interval exchanges and some Toeplitz sequences). In this talk I am going to present results concerning the number of automorphisms and factors of systems of finite topological rank, as well as closure properties of this class with respect to factors and related combinatorial operations.

Shigeki Akiyama (University of Tsukuba)
2021-06-08 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/48/
Title: Counting balanced words and related problems (https://researchseminars.org/talk/OWNS/48/)
by Shigeki Akiyama (University of Tsukuba) as part of One World Numeration seminar

Abstract
Balanced words and Sturmian words are ubiquitous and appear in the intersection of many areas of mathematics. In this talk, I try to explain an idea of S. Yasutomi to study finite balanced words. His method gives a nice way to enumerate number of balanced words of given length, slope and intercept. Applying this idea, we can obtain precise asymptotic formula for balanced words. The result is connected to some classical topics in number theory, such as Farey fraction, Riemann Hypothesis and Large sieve inequality.

Sam Chow (University of Warwick)
2021-06-15 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/49/
Title: Dyadic approximation in the Cantor set (https://researchseminars.org/talk/OWNS/49/)
by Sam Chow (University of Warwick) as part of One World Numeration seminar

Abstract
We investigate the approximation rate of a typical element of the Cantor set by dyadic rationals. This is a manifestation of the times-two-times-three phenomenon, and is joint work with Demi Allen and Han Yu.

Lingmin Liao (Université Paris-Est Créteil Val de Marne)
2021-06-22 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/50/
Title: Simultaneous Diophantine approximation of the orbits of the dynamical systems x2 and x3 (https://researchseminars.org/talk/OWNS/50/)
by Lingmin Liao (Université Paris-Est Créteil Val de Marne) as part of One World Numeration seminar

Abstract
We study the sets of points whose orbits of the dynamical systems x2 and x3 simultaneously approach to a given point, with a given speed. A zero-one law for the Lebesgue measure of such sets is established. The Hausdorff dimensions are also determined for some special speeds. One dimensional formula among them is established under the abc conjecture. At the same time, we also study the Diophantine approximation of the orbits of a diagonal matrix transformation of a torus, for which the properties of the (negative) beta transformations are involved. This is a joint work with Bing Li, Sanju Velani and Evgeniy Zorin.

Polina Vytnova (University of Warwick)
2021-06-29 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/51/
Title: Hausdorff dimension of Gauss-Cantor sets and their applications to the study of classical Markov spectrum (https://researchseminars.org/talk/OWNS/51/)
by Polina Vytnova (University of Warwick) as part of One World Numeration seminar

Abstract
The classical Lagrange and Markov spectra are subsets of the real line which arise in connection with some problems in theory Diophantine approximation theory. In 1921 O. Perron gave a definition in terms of continued fractions, which allowed to study the Markov and Lagrange spectra using limit sets of iterated function schemes.

In this talk we will see how the first transition point, where the Markov spectra acquires the full measure can be computed by the means of estimating Hausdorff dimension of the certain Gauss-Cantor sets.

The talk is based on a joint work with C. Matheus, C. G. Moreira and M. Pollicott.

Niclas Technau (University of Wisconsin - Madison)
2021-07-06 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/52/
Title: Littlewood and Duffin-Schaeffer-type problems in diophantine approximation (https://researchseminars.org/talk/OWNS/52/)
by Niclas Technau (University of Wisconsin - Madison) as part of One World Numeration seminar

Abstract
Gallagher's theorem describes the multiplicative diophantine approximation rate of a typical vector. Recently Sam Chow and I establish a fully-inhomogeneous version of Gallagher's theorem, and a diophantine fibre refinement. In this talk I outline the proof, and the tools involved in it.

Oleg Karpenkov (University of Liverpool)
2021-09-07 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/53/
Title: On Hermite's problem, Jacobi-Perron type algorithms, and Dirichlet groups (https://researchseminars.org/talk/OWNS/53/)
by Oleg Karpenkov (University of Liverpool) as part of One World Numeration seminar

Abstract
In this talk we introduce a new modification of the Jacobi-Perron algorithm in the three dimensional case. This algorithm is periodic for the case of totally-real conjugate cubic vectors. To the best of our knowledge this is the first Jacobi-Perron type algorithm for which the cubic periodicity is proven. This provides an answer in the totally-real case to the question of algebraic periodicity for cubic irrationalities posed in 1848 by Ch.Hermite.

We will briefly discuss a new approach which is based on geometry of numbers. In addition we point out one important application of Jacobi-Perron type algorithms to the computation of independent elements in the maximal groups of commuting matrices of algebraic irrationalities.

Henna Koivusalo (University of Vienna)
2020-06-02 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/54/
Title: Linear repetition in polytopal cut and project sets (https://researchseminars.org/talk/OWNS/54/)
by Henna Koivusalo (University of Vienna) as part of One World Numeration seminar

Abstract
Cut and project sets are aperiodic point patterns obtained by projecting an irrational slice of the integer lattice to a subspace. One way of classifying aperiodic sets is to study repetition of finite patterns, where sets with linear pattern repetition can be considered as the most ordered aperiodic sets.
Repetitivity of a cut and project set depends on the slope and shape of the irrational slice. The cross-section of the slice is known as the window. In an earlier work it was shown that for cut and project sets with a cube window, linear repetitivity holds if and only if the following two conditions are satisfied: (i) the set has minimal complexity and (ii) the irrational slope satisfies a certain Diophantine condition. In a new joint work with Jamie Walton, we give a generalisation of this result for other polytopal windows, under mild geometric conditions. A key step in the proof is a decomposition of the cut and project scheme, which allows us to make sense of condition (ii) for general polytopal windows.

Steve Jackson (University of North Texas)
2021-09-14 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/55/
Title: Descriptive complexity in numeration systems (https://researchseminars.org/talk/OWNS/55/)
by Steve Jackson (University of North Texas) as part of One World Numeration seminar

Abstract
Descriptive set theory gives a means of calibrating the complexity of sets, and we focus on some sets occurring in numerations systems. Also, the descriptive complexity of the difference of two sets gives a notion of the logical independence of the sets. A classic result of Ki and Linton says that the set of normal numbers for a given base is a $\boldsymbol{\Pi}^0_3$ complete set. In work with Airey, Kwietniak, and Mance we extend to other numerations systems such as continued fractions, $\beta$-expansions, and GLS expansions. In work with Mance and Vandehey we show that the numbers which are continued fraction normal but not base $b$ normal is complete at the expected level of $D_2(\boldsymbol{\Pi}^0_3)$. An immediate corollary is that this set is uncountable, a result (due to Vandehey) only known previously assuming the generalized Riemann hypothesis.

Maria Siskaki (University of Illinois at Urbana-Champaign)
2021-09-21 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/56/
Title: The distribution of reduced quadratic irrationals arising from continued fraction expansions (https://researchseminars.org/talk/OWNS/56/)
by Maria Siskaki (University of Illinois at Urbana-Champaign) as part of One World Numeration seminar

Abstract
It is known that the reduced quadratic irrationals arising from regular continued fraction expansions are uniformly distributed when ordered by their length with respect to the Gauss measure. In this talk, I will describe a number theoretical approach developed by Kallies, Ozluk, Peter and Snyder, and then by Boca, that gives the error in the asymptotic behavior of this distribution. Moreover, I will present the respective result for the distribution of reduced quadratic irrationals that arise from even (joint work with F. Boca) and odd continued fractions.

Philipp Hieronymi (Universität Bonn)
2021-09-28 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/57/
Title: A strong version of Cobham's theorem (https://researchseminars.org/talk/OWNS/57/)
by Philipp Hieronymi (Universität Bonn) as part of One World Numeration seminar

Abstract
Let $k,l>1$ be two multiplicatively independent integers. A subset $X$ of $\mathbb{N}^n$ is $k$-recognizable if the set of $k$-ary representations of $X$ is recognized by some finite automaton. Cobham’s famous theorem states that a subset of the natural numbers is both $k$-recognizable and $l$-recognizable if and only if it is Presburger-definable (or equivalently: semilinear). We show the following strengthening. Let $X$ be $k$-recognizable, let $Y$ be $l$-recognizable such that both $X$ and $Y$ are not Presburger-definable. Then the first-order logical theory of $(\mathbb{N},+,X,Y)$ is undecidable. This is in contrast to a well-known theorem of Büchi that the first-order logical theory of $(\mathbb{N},+,X)$ is decidable. Our work strengthens and depends on earlier work of Villemaire and Bès.

The essence of Cobham's theorem is that recognizability depends strongly on the choice of the base $k$. Our results strengthens this: two non-Presburger definable sets that are recognizable in multiplicatively independent bases, are not only distinct, but together computationally intractable over Presburger arithmetic.

This is joint work with Christian Schulz.

Lulu Fang (Nanjing University of Science and Technology)
2021-10-05 12:30 - 13:00 | https://researchseminars.org/talk/OWNS/58/
Title: On upper and lower fast Khintchine spectra in continued fractions (https://researchseminars.org/talk/OWNS/58/)
by Lulu Fang (Nanjing University of Science and Technology) as part of One World Numeration seminar

Abstract
Let $\psi:\mathbb{N}\to \mathbb{R}^+$ be a function satisfying $\psi(n)/n\to \infty$ as $n \to \infty$.
We investigate from a multifractal analysis point of view the growth speed of the sums $\sum^n_{k=1}\log a_k(x)$
with respect to $\psi(n)$, where $x=[a_1(x),a_2(x),\cdots]$ denotes the continued fraction expansion of $x\in (0,1)$.
The (upper, lower) fast Khintchine spectrum is defined as the Hausdorff dimension of the set of points $x\in(0,1)$
for which the (upper, lower) limit of $\frac{1}{\psi(n)}\sum^n_{k=1}\log a_k(x)$ is equal to $1$. These three spectra
have been studied by Fan, Liao ,Wang \& Wu (2013, 2016), Liao \& Rams (2016). In this talk, we will give a new look
at the fast Khintchine spectrum, and provide a full description of upper and lower fast Khintchine spectra. The latter
improves a result of Liao and Rams (2016).

Liangang Ma (Binzhou University)
2021-10-12 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/59/
Title: Inflection points in the Lyapunov spectrum for IFS on intervals (https://researchseminars.org/talk/OWNS/59/)
by Liangang Ma (Binzhou University) as part of One World Numeration seminar

Abstract: TBA

Taylor Jones (University of North Texas)
2021-10-05 13:00 - 13:30 | https://researchseminars.org/talk/OWNS/60/
Title: On the Existence of Numbers with Matching Continued Fraction and Decimal Expansion (https://researchseminars.org/talk/OWNS/60/)
by Taylor Jones (University of North Texas) as part of One World Numeration seminar

Abstract
A Trott number in base 10 is one whose continued fraction expansion agrees with its base 10 expansion in the sense that $[0;a_1,a_2,\dots] = 0.(a_1)(a_2) \cdots$ where $(a_i)$ represents the string of digits of $a_i$. As an example $[0;3,29,54,7,\dots] = 0.329547\cdots$.
An analogous definition may be given for a Trott number in any integer base $b>1$, the set of which we denote by $T_b$. The first natural question is whether $T_b$ is empty, and if not, for which $b$? We discuss the history of the problem, and give a heuristic process for constructing such numbers. We show that $T_{10}$ is indeed non-empty, and uncountable. With more delicate techniques, a complete classification may be given to all $b$ for which $T_b$ is non-empty. We also discuss some further results, such as a (non-trivial) upper bound on the Hausdorff dimension of $T_b$, as well as the question of whether the intersection of $T_b$ and $T_c$ can be non-empty.

Mélodie Lapointe (Université de Paris)
2021-10-19 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/61/
Title: q-analog of the Markoff injectivity conjecture (https://researchseminars.org/talk/OWNS/61/)
by Mélodie Lapointe (Université de Paris) as part of One World Numeration seminar

Abstract
The Markoff injectivity conjecture states that $w\mapsto\mu(w)_{12}$ is injective on the set of Christoffel words where $\mu:\{\mathtt{0},\mathtt{1}\}^*\to\mathrm{SL}_2(\mathbb{Z})$ is a certain homomorphism and $M_{12}$ is the entry above the diagonal of a $2\times2$ matrix $M$. Recently, Leclere and Morier-Genoud (2021) proposed a $q$-analog $\mu_q$ of $\mu$ such that $\mu_{q\to1}(w)_{12}=\mu(w)_{12}$ is the Markoff number associated to the Christoffel word $w$. We show that there exists an order $<_{radix}$ on $\{\mathtt{0},\mathtt{1}\}^*$ such that for every balanced sequence $s \in \{\mathtt{0},\mathtt{1}\}^\mathbb{Z}$ and for all factors $u, v$ in the language of $s$ with $u <_{radix} v$, the difference $\mu_q(v)_{12} - \mu_q(u)_{12}$ is a nonzero polynomial of indeterminate $q$ with nonnegative integer coefficients. Therefore, for every $q>0$, the map $\{\mathtt{0},\mathtt{1}\}^*\to\mathbb{R}$ defined by $w\mapsto\mu_q(w)_{12}$ is increasing thus injective over the language of a balanced sequence. The proof uses an equivalence between balanced sequences satisfying some Markoff property and indistinguishable asymptotic pairs.

Michael Baake (Universität Bielefeld)
2021-10-26 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/62/
Title: Spectral aspects of aperiodic dynamical systems (https://researchseminars.org/talk/OWNS/62/)
by Michael Baake (Universität Bielefeld) as part of One World Numeration seminar

Abstract
One way to analyse aperiodic systems employs spectral notions, either via dynamical systems theory or via harmonic analysis. In this talk, we will look at two particular aspects of this, after a quick overview of how the diffraction measure can be used for this purpose. First, we consider some concequences of inflation rules on the spectra via renormalisation, and how to use it to exclude absolutely continuous componenta. Second, we take a look at a class of dynamical systems of number-theoretic origin, how they fit into the spectral picture, and what (other) methods there are to distinguish them.

Pieter Allaart (University of North Texas)
2021-11-02 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/63/
Title: On the existence of Trott numbers relative to multiple bases (https://researchseminars.org/talk/OWNS/63/)
by Pieter Allaart (University of North Texas) as part of One World Numeration seminar

Abstract
Trott numbers are real numbers in the interval $(0,1)$ whose continued fraction expansion equals their base-$b$ expansion, in a certain liberal but natural sense. They exist in some bases, but not in all. In a previous OWNS talk, T. Jones sketched a proof of the existence of Trott numbers in base 10. In this talk I will discuss some further properties of these Trott numbers, and focus on the question: Can a number ever be Trott in more than one base at once? While the answer is almost certainly "no", a full proof of this seems currently out of reach. But we obtain some interesting partial answers by using a deep theorem from Diophantine approximation.

Zhiqiang Wang (East China Normal University)
2021-11-09 13:30 - 14:00 | https://researchseminars.org/talk/OWNS/64/
Title: How inhomogeneous Cantor sets can pass a point (https://researchseminars.org/talk/OWNS/64/)
by Zhiqiang Wang (East China Normal University) as part of One World Numeration seminar

Abstract
Abstract: For $x > 0$, we define $$\Upsilon(x) = \{ (a,b): x\in E_{a,b}, a>0, b>0, a+b \le 1 \},$$ where the set $E_{a,b}$ is the unique nonempty compact invariant set generated by the inhomogeneous IFS $$\{ f_0(x) = a x, f_1(x) = b(x+1) \}.$$ We show the set $\Upsilon(x)$ is a Lebesgue null set with full Hausdorff dimension in $\mathbb{R}^2$, and the intersection of sets $\Upsilon(x_1), \Upsilon(x_2), \dots, \Upsilon(x_\ell)$ still has full Hausdorff dimension $\mathbb{R}^2$ for any finitely many positive real numbers $x_1, x_2, \dots, x_\ell$.

Lucía Rossi (Montanuniversität Leoben)
2021-11-16 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/65/
Title: Rational self-affine tiles associated to (nonstandard) digit systems (https://researchseminars.org/talk/OWNS/65/)
by Lucía Rossi (Montanuniversität Leoben) as part of One World Numeration seminar

Abstract
In this talk we will introduce the notion of rational self-affine tiles, which are fractal-like sets that arise as the solution of a set equation associated to a digit system that consists of a base, given by an expanding rational matrix, and a digit set, given by vectors. They can be interpreted as the set of “fractional parts” of this digit system, and the challenge of this theory is that these sets do not live in a Euclidean space, but on more general spaces defined in terms of Laurent series. Steiner and Thuswaldner defined rational self-affine tiles for the case where the base is a rational matrix with irreducible characteristic polynomial. We present some tiling results that generalize the ones obtained by Lagarias and Wang: we consider arbitrary expanding rational matrices as bases, and simultaneously allow the digit sets to be nonstandard (meaning they are not a complete set of residues modulo the base). We also state some topological properties of rational self-affine tiles and give a criterion to guarantee positive measure in terms of the digit set.

Sascha Troscheit (Universität Wien)
2021-11-23 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/66/
Title: Analogues of Khintchine's theorem for random attractors (https://researchseminars.org/talk/OWNS/66/)
by Sascha Troscheit (Universität Wien) as part of One World Numeration seminar

Abstract
Khintchine’s theorem is an important result in number theory which links the Lebesgue measure of certain limsup sets with the convergence/divergence of naturally occurring volume sums. This behaviour has been observed for deterministic fractal sets and inspired by this we investigate the random settings. Introducing randomisation into the problem makes some parts more tractable, while posing separate new challenges. In this talk, I will present joint work with Simon Baker where we provide sufficient conditions for a large class of stochastically self-similar and self-affine attractors to have positive Lebesgue measure.

Jamie Walton (University of Glasgow)
2021-12-07 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/67/
Title: Extending the theory of symbolic substitutions to compact alphabets (https://researchseminars.org/talk/OWNS/67/)
by Jamie Walton (University of Glasgow) as part of One World Numeration seminar

Abstract
In this work, joint with Neil Mañibo and Dan Rust, we consider an extension of the theory of symbolic substitutions to infinite alphabets, by requiring the alphabet to carry a compact, Hausdorff topology for which the substitution is continuous. Such substitutions have been considered before, in particular by Durand, Ormes and Petite for zero-dimensional alphabets, and Queffélec in the constant length case. We find a simple condition which ensures that an associated substitution operator is quasi-compact, which we conjecture to always be satisfied for primitive substitutions on countable alphabets. In the primitive case this implies the existence of a unique natural tile length function and, for a recognisable substitution, that the associated shift space is uniquely ergodic. The main tools come from the theory of positive operators on Banach spaces. Very few prerequisites will be assumed, and the theory will be demonstrated via examples.

Younès Tierce (Université de Rouen Normandie)
2021-11-09 14:00 - 14:30 | https://researchseminars.org/talk/OWNS/68/
Title: Extensions of the random beta-transformation (https://researchseminars.org/talk/OWNS/68/)
by Younès Tierce (Université de Rouen Normandie) as part of One World Numeration seminar

Abstract
Let $\beta \in (1,2)$ and $I_\beta := [0,\frac{1}{\beta-1}]$. Almost every real number of $I_\beta$ has infinitely many expansions in base $\beta$, and the random $\beta$-transformation generates all these expansions. We present the construction of a "geometrico-symbolic" extension of the random $\beta$-transformation, providing a new proof of the existence and unicity of an absolutely continuous invariant probability measure, and an expression of the density of this measure. This extension shows off some nice renewal times, and we use these to prove that the natural extension of the system is a Bernoulli automorphism.

Valérie Berthé, Pierre Arnoux, ...
2021-12-14 13:00 - 15:00 | https://researchseminars.org/talk/OWNS/69/
Title: Special session commemorating Shunji Ito (1943-2021) (https://researchseminars.org/talk/OWNS/69/)
by Valérie Berthé, Pierre Arnoux, ... as part of One World Numeration seminar

Abstract
Introduction by Pierre Arnoux, short talk by Valérie Berthé, contributions by Maki Furukado, Cor Kraaikamp, Hui Rao, Robbie Robinson, Shin'Ichi Yasutomi, Shigeki Akiyama, and Hiromi Ei.

Fan Lü (Sichuan Normal University)
2021-12-21 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/70/
Title: Multiplicative Diophantine approximation in the parameter space of beta-dynamical system (https://researchseminars.org/talk/OWNS/70/)
by Fan Lü (Sichuan Normal University) as part of One World Numeration seminar

Abstract
Beta-transformation is a special kind of expanding dynamics, the total information of which can be determined by the orbits of some critical points (e.g., the point 1). Letting $T_{\beta}$ be the beta-transformation with $\beta>1$ and $x$ be a fixed point in $(0,1]$, we consider the set of parameters $(\alpha, \beta)$, such that the multiple $\|T^n_{\alpha}(x)\|\|T^n_{\beta}(x)\|$ is well approximated or badly approximated. The Gallagher-type question, Jarník-type question as well as the badly approximable pairs, i.e., Littlewood-type question are studied in detail.

Philipp Gohlke (Universität Bielefeld)
2022-01-11 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/71/
Title: Zero measure spectrum for multi-frequency Schrödinger operators (https://researchseminars.org/talk/OWNS/71/)
by Philipp Gohlke (Universität Bielefeld) as part of One World Numeration seminar

Abstract
Cantor spectrum of zero Lebesgue measure is a striking feature of Schrödinger operators associated with certain models of aperiodic order, like primitive substitution systems or Sturmian subshifts. This is known to follow from a condition introduced by Boshernitzan that establishes that on infinitely many scales words of the same length appear with a similar frequency. Building on works of Berthé–Steiner–Thuswaldner and Fogg–Nous we show that on the two-dimensional torus, Lebesgue almost every translation admits a natural coding such that the associated subshift satisfies the Boshernitzan criterion (joint work with J.Chaika, D.Damanik and J.Fillman).

Agamemnon Zafeiropoulos (NTNU)
2022-01-18 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/72/
Title: The order of magnitude of Sudler products (https://researchseminars.org/talk/OWNS/72/)
by Agamemnon Zafeiropoulos (NTNU) as part of One World Numeration seminar

Abstract
Given an irrational $\alpha \in [0,1] \smallsetminus \mathbb{Q}$, we define the corresponding Sudler product by $$ P_N(\alpha) = \prod_{n=1}^{N}2|\sin (\pi n \alpha)|. $$ In joint work with C. Aistleitner and N. Technau, we show that when $\alpha = [0;b,b,b…]$ is a quadratic irrational with all partial quotients in its continued fraction expansion equal to some integer b, the following hold:

- If $b\leq 5$, then $\liminf_{N\to \infty}P_N(\alpha) >0$ and $\limsup_{N\to \infty} P_N(\alpha)/N < \infty$.

-If $b\geq 6$, then $\liminf_{N\to \infty}P_N(\alpha) = 0$ and $\limsup_{N\to \infty} P_N(\alpha)/N = \infty$.

We also present an analogue of the previous result for arbitrary quadratic irrationals (joint work with S. Grepstad and M. Neumueller).

Claudio Bonanno (Università di Pisa)
2022-01-25 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/73/
Title: Infinite ergodic theory and a tree of rational pairs (https://researchseminars.org/talk/OWNS/73/)
by Claudio Bonanno (Università di Pisa) as part of One World Numeration seminar

Abstract
The study of the continued fraction expansions of real numbers by ergodic methods is now a classical and well-known part of the theory of dynamical systems. Less is known for the multi-dimensional expansions. I will present an ergodic approach to a two-dimensional continued fraction algorithm introduced by T. Garrity, and show how to get a complete tree of rational pairs by using the Farey sum of fractions. The talk is based on joint work with A. Del Vigna and S. Munday.

Magdaléna Tinková (Czech Technical University in Prague)
2022-02-08 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/74/
Title: Universal quadratic forms, small norms and traces in families of number fields (https://researchseminars.org/talk/OWNS/74/)
by Magdaléna Tinková (Czech Technical University in Prague) as part of One World Numeration seminar

Abstract
In this talk, we will discuss universal quadratic forms over number fields and their connection with additively indecomposable integers. In particular, we will focus on Shanks' family of the simplest cubic fields. This is joint work with Vítězslav Kala.

Michael Coons (Universität Bielefeld)
2022-03-08 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/75/
Title: A spectral theory of regular sequences (https://researchseminars.org/talk/OWNS/75/)
by Michael Coons (Universität Bielefeld) as part of One World Numeration seminar

Abstract
A few years ago, Michael Baake and I introduced a probability measure associated to Stern’s diatomic sequence, an example of a regular sequence—sequences which generalise constant length substitutions to infinite alphabets. In this talk, I will discuss extensions of these results to more general regular sequences as well as further properties of these measures. This is joint work with several people, including Michael Baake, James Evans, Zachary Groth and Neil Manibo.

Jonas Jankauskas (Vilnius University)
2022-02-01 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/76/
Title: Digit systems with rational base matrix over lattices (https://researchseminars.org/talk/OWNS/76/)
by Jonas Jankauskas (Vilnius University) as part of One World Numeration seminar

Abstract
Let $A$ be a matrix with rational entries and no eigenvalue in absolute value smaller than 1. Let $\mathbb{Z}^d[A]$ be the minimal $A$-invariant $\mathbb{Z}$-module, generated by integer vectors and the matrix $A$. In 2018, we have shown that one can find a finite set $D$ of vectors, such that each element of $\mathbb{Z}^d[A]$ has a finite radix expansion in base $A$ using only the digits from $D$, i.e. $\mathbb{Z}^d[A]=D[A]$. This is called 'the finiteness property' of a digit system. In the present talk I will review more recent developments in mathematical machinery, that enable us to build finite digit systems over lattices using reasonably small digit sets, and even to do some practical computations with them on a computer. Tools that we use are the generalized rotation bases with digit sets that have 'good' convex properties, the semi-direct ('twisted') sums of such rotational digit systems, and the special, 'restricted' version of the remainder division that preserves the lattice $\mathbb{Z}^d$ and can be extended to $\mathbb{Z}^d[A]$. This is joint work with J. Thuswaldner, "Rational Matrix Digit Systems", to appear in "Linear and Multilinear Algebra".

Wolfgang Steiner (CNRS, Université de Paris)
2022-02-15 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/77/
Title: Unique double base expansions (https://researchseminars.org/talk/OWNS/77/)
by Wolfgang Steiner (CNRS, Université de Paris) as part of One World Numeration seminar

Abstract
For pairs of real bases $\beta_0, \beta_1 > 1$, we study expansions of the form
$\sum_{k=1}^\infty i_k / (\beta_{i_1} \beta_{i_2} \cdots \beta_{i_k})$
with digits $i_k \in \{0,1\}$.
We characterise the pairs admitting non-trivial unique expansions as well as those admitting uncountably many unique expansions, extending recent results of Neunhäuserer (2021) and Zou, Komornik and Lu (2021).
Similarly to the study of unique $\beta$-expansions with three digits by the speaker (2020), this boils down to determining the cardinality of binary shifts defined by lexicographic inequalities.
Labarca and Moreira (2006) characterised when such a shift is empty, at most countable or uncountable, depending on the position of the lower and upper bounds with respect to Thue-Morse-Sturmian words.

This is joint work with Vilmos Komornik and Yuru Zou.

Daniel Krenn (Universität Salzburg)
2022-03-01 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/78/
Title: $k$-regular sequences: Asymptotics and Decidability (https://researchseminars.org/talk/OWNS/78/)
by Daniel Krenn (Universität Salzburg) as part of One World Numeration seminar

Abstract
A sequence $x(n)$ is called $k$-regular, if the set of subsequences $x(k^j n + r)$ is contained in a finitely generated module. In this talk, we will consider the asymptotic growth of $k$-regular sequences. When is it possible to compute it? ...and when not? If possible, how precisely can we compute it? If not, is it just a lack of methods or are the underlying decision questions recursively solvable (i.e., decidable in a computational sense)? We will discuss answers to these questions. To round off the picture, we will consider further decidability questions around $k$-regular sequences and the subclass of $k$-automatic sequences.

This is based on joint works with Clemens Heuberger and with Jeffrey Shallit.

Pierre Popoli (Université de Lorraine)
2022-03-15 13:30 - 14:30 | https://researchseminars.org/talk/OWNS/79/
Title: Maximum order complexity for some automatic and morphic sequences along polynomial values (https://researchseminars.org/talk/OWNS/79/)
by Pierre Popoli (Université de Lorraine) as part of One World Numeration seminar

Abstract
Automatic sequences are not suitable sequences for cryptographic applications since both their subword complexity and their expansion complexity are small, and their correlation measure of order 2 is large. These sequences are highly predictable despite having a large maximum order complexity. However, recent results show that polynomial subsequences of automatic sequences, such as the Thue-Morse sequence or the Rudin-Shapiro sequence, are better candidates for pseudorandom sequences. A natural generalization of automatic sequences are morphic sequences, given by a fixed point of a prolongeable morphism that is not necessarily uniform. In this talk, I will present my results on lowers bounds for the maximum order complexity of the Thue-Morse sequence, the Rudin-Shapiro sequence and the sum of digits function in Zeckendorf base, which are respectively automatics and morphic sequences.

Tingyu Zhang (East China Normal University)
2022-03-29 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/80/
Title: Random $\beta$-transformation on fat Sierpiński gasket (https://researchseminars.org/talk/OWNS/80/)
by Tingyu Zhang (East China Normal University) as part of One World Numeration seminar

Abstract
We define the notions of greedy, lazy and random transformations on fat Sierpiński gasket. We determine the bases, for which the system has a unique measure of maximal entropy and an invariant measure of product type, with one coordinate being absolutely continuous with respect to Lebesgue measure.

This is joint work with K. Dajani and W. Li.

Jungwon Lee (University of Warwick)
2022-04-05 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/81/
Title: Dynamics of Ostrowski skew-product: Limit laws and Hausdorff dimensions (https://researchseminars.org/talk/OWNS/81/)
by Jungwon Lee (University of Warwick) as part of One World Numeration seminar

Abstract
We discuss a dynamical study of the Ostrowski skew-product map in the context of inhomogeneous Diophantine approximation. We plan to outline the setup/ strategy based on transfer operator analysis and applications in arithmetic of number fields (joint with Valérie Berthé).

Eda Cesaratto (Univ. Nac. de Gral. Sarmiento & CONICET)
2022-04-12 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/82/
Title: Lochs-type theorems beyond positive entropy (https://researchseminars.org/talk/OWNS/82/)
by Eda Cesaratto (Univ. Nac. de Gral. Sarmiento & CONICET) as part of One World Numeration seminar

Abstract
Lochs' theorem and its generalizations are conversion theorems that relate the number of digits determined in one expansion of a real number as a function of the number of digits given in some other expansion. In its original version, Lochs' theorem related decimal expansions with continued fraction expansions. Such conversion results can also be stated for sequences of interval partitions under suitable assumptions, with results holding almost everywhere, or in measure, involving the entropy. This is the viewpoint we develop here. In order to deal with sequences of partitions beyond positive entropy, this paper introduces the notion of log-balanced sequences of partitions, together with their weight functions. These are sequences of interval partitions such that the logarithms of the measures of their intervals at each depth are roughly the same. We then state Lochs-type theorems which work even in the case of zero entropy, in particular for several important log-balanced sequences of partitions of a number-theoretic nature.

This is joint work with Valérie Berthé (IRIF), Pablo Rotondo (U. Gustave Eiffel) and Martín Safe (Univ. Nac. del Sur & CONICET, Argentina).

Paulina Cecchi Bernales (Universidad de Chile)
2022-04-19 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/83/
Title: Coboundaries and eigenvalues of finitary S-adic systems (https://researchseminars.org/talk/OWNS/83/)
by Paulina Cecchi Bernales (Universidad de Chile) as part of One World Numeration seminar

Abstract
An S-adic system is a shift space obtained by performing an infinite composition of morphisms defined over possibly different finite alphabets. It is said to be finitary if these morphisms are taken from a finite set. S-adic systems are a generalization of substitution shifts. In this talk we will discuss spectral properties of finitary S-adic systems. Our departure point will be a theorem by B. Host which characterizes eigenvalues of substitution shifts, and where coboundaries appear as a key tool. We will introduce the notion of S-adic coboundaries and present some results which show how they are related with eigenvalues of S-adic systems. We will also present some applications of our results to constant-length finitary S-adic systems.

This is joint work with Valérie Berthé and Reem Yassawi.

Nicolas Chevallier (Université de Haute Alsace)
2022-05-03 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/84/
Title: Best Diophantine approximations in the complex plane with Gaussian integers (https://researchseminars.org/talk/OWNS/84/)
by Nicolas Chevallier (Université de Haute Alsace) as part of One World Numeration seminar

Abstract
Starting with the minimal vectors in lattices over Gaussian integers in $\C^2$, we define a algorithm that finds the sequence of minimal vectors of any unimodular lattice in $\C^2$.
Restricted to lattices associated with complex numbers this algorithm find all the best Diophantine approximations of a complex numbers.
Following Doeblin, Lenstra, Bosma, Jager and Wiedijk, we study the limit distribution of the sequence of products $(u_{n1}u_{n2})_n$ where $(u _n=( u_{n1},u_{n2} ))_n$ is the sequence of minimal vectors of a lattice in $C^2$. We show that there exists a measure in $\C$ which is the limit distribution of the sequence of products of almost all unimodular lattices.

Vilmos Komornik (Shenzhen University and Université de Strasbourg)
2022-05-17 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/85/
Title: Topology of univoque sets in real base expansions (https://researchseminars.org/talk/OWNS/85/)
by Vilmos Komornik (Shenzhen University and Université de Strasbourg) as part of One World Numeration seminar

Abstract
We report on a recent joint paper with Martijn de Vries and Paola Loreti. Given a positive integer $M$ and a real number $1 < q\le M+1$, an expansion of a real number $x \in \left[0,M/(q-1)\right]$ over the alphabet $A=\{0,1,\ldots,M\}$ is a sequence $(c_i) \in A^{\mathbb{N}}$ such that $x=\sum_{i=1}^{\infty}c_iq^{-i}$. Generalizing many earlier results, we investigate the topological properties of the set $U_q$ consisting of numbers $x$ having a unique expansion of this form, and the combinatorial properties of the set $U_q'$ consisting of their corresponding expansions.

Émilie Charlier (Université de Liège)
2022-05-24 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/86/
Title: Spectrum, algebraicity and normalization in alternate bases (https://researchseminars.org/talk/OWNS/86/)
by Émilie Charlier (Université de Liège) as part of One World Numeration seminar

Abstract
The first aim of this work is to give information about the algebraic properties of alternate bases determining sofic systems. We exhibit two conditions: one necessary and one sufficient. Comparing the setting of alternate bases to that of one real base, these conditions exhibit a new phenomenon: the bases should be expressible as rational functions of their product. The second aim is to provide an analogue of Frougny's result concerning normalization of real bases representations. Under some suitable condition (i.e., our previous sufficient condition for being a sofic system), we prove that the normalization function is computable by a finite Büchi automaton, and furthermore, we effectively construct such an automaton. An important tool in our study is the spectrum of numeration systems associated with alternate bases. For our purposes, we use a generalized concept of spectrum associated with a complex base and complex digits, and we study its topological properties.

This is joint work with Célia Cisternino, Zuzana Masáková and Edita Pelantová.

Verónica Becher (Universidad de Buenos Aires & CONICET Argentina)
2022-05-31 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/87/
Title: Poisson generic real numbers (https://researchseminars.org/talk/OWNS/87/)
by Verónica Becher (Universidad de Buenos Aires & CONICET Argentina) as part of One World Numeration seminar

Abstract
Years ago Zeev Rudnick defined the Poisson generic real numbers as those where the number of occurrences of the long strings in the initial segments of their fractional expansions in some base have the Poisson distribution. Yuval Peres and Benjamin Weiss proved that almost all real numbers, with respect to Lebesgue measure, are Poisson generic. They also showed that Poisson genericity implies Borel normality but the two notions do not coincide, witnessed by the famous Champernowne constant. We recently showed that there are computable Poisson generic real numbers and that all Martin-Löf real numbers are Poisson generic.
This is joint work Nicolás Álvarez and Martín Mereb.

Sophie Morier-Genoud (Université Reims Champagne Ardenne)
2022-06-07 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/88/
Title: q-analogues of real numbers (https://researchseminars.org/talk/OWNS/88/)
by Sophie Morier-Genoud (Université Reims Champagne Ardenne) as part of One World Numeration seminar

Abstract
Classical sequences of numbers often lead to interesting q-analogues. The most popular among them are certainly the q-integers and the q-binomial coefficients which both appear in various areas of mathematics and physics. With Valentin Ovsienko we recently suggested a notion of q-rationals based on combinatorial properties and continued fraction expansions. The definition of q-rationals naturally extends the one of q-integers and leads to a ratio of polynomials with positive integer coefficients. I will explain the construction and give the main properties. In particular I will briefly mention connections with the combinatorics of posets, cluster algebras, Jones polynomials, homological algebra. Finally I will also present further developments of the theory, leading to the notion of q-irrationals and q-unimodular matrices.

James A. Yorke (University of Maryland)
2022-06-21 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/89/
Title: Large and Small Chaos Models (https://researchseminars.org/talk/OWNS/89/)
by James A. Yorke (University of Maryland) as part of One World Numeration seminar

Abstract
To set the scene, I will discuss one large model, a whole-Earth model for predicting the weather, and how to initialize such a model and what aspects of chaos are essential. Then I will discuss a couple related “very simple” maps that tell us a great deal about very complex models. The results on simple models are new. I will discuss the logistic map mx(1-x). Its dynamics can make us rethink climate models. Also, we have created a piecewise linear map on a 3D cube that is unstable in 2 dimensions in some places and unstable in 1 in others. It has a dense set of periodic points that are 1 D unstable and another dense set of periodic points that are all 2 D unstable. I will also discuss a new project whose tentative title is “ Can the flap of butterfly's wings shift a tornado out of Texas -- without chaos?

Charlene Kalle (Universiteit Leiden)
2022-07-05 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/90/
Title: Random Lüroth expansions (https://researchseminars.org/talk/OWNS/90/)
by Charlene Kalle (Universiteit Leiden) as part of One World Numeration seminar

Abstract
Since the introduction of Lüroth expansions by Lüroth in his paper from 1883 many results have appeared on their approximation properties. In 1990 Kalpazidou, Knopfmacher and Knopfmacher introduced alternating Lüroth expansions and studied their properties. A comparison between the two and other comparable number systems was then given by Barrionuevo, Burton, Dajani and Kraaikamp in 1996. In this talk we introduce a family of random dynamical systems that produce many Lüroth type expansions at once. Topics that we consider are periodic expansions, universal expansions, speed of convergence and approximation coefficients. This talk is based on joint work with Marta Maggioni.

Ruofan Li (South China University of Technology)
2022-07-12 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/91/
Title: Rational numbers in $\times b$-invariant sets (https://researchseminars.org/talk/OWNS/91/)
by Ruofan Li (South China University of Technology) as part of One World Numeration seminar

Abstract
Let $b \ge 2$ be an integer and $S$ be a finite non-empty set of primes not containing divisors of $b$. For any $\times b$-invariant, non-dense subset $A$ of $[0,1)$, we prove the finiteness of rational numbers in $A$ whose denominators can only be divided by primes in $S$. A quantitative result on the largest prime divisors of the denominators of rational numbers in $A$ is also obtained.
This is joint work with Bing Li and Yufeng Wu.

Benedict Sewell (Alfréd Rényi Institute)
2022-09-13 12:30 - 13:30 | https://researchseminars.org/talk/OWNS/92/
Title: An upper bound on the box-counting dimension of the Rauzy gasket (https://researchseminars.org/talk/OWNS/92/)
by Benedict Sewell (Alfréd Rényi Institute) as part of One World Numeration seminar

Abstract: TBA

Niels Langeveld (Montanuniversität Leoben)
2022-09-27 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/93/
Title: $N$-continued fractions and $S$-adic sequences (https://researchseminars.org/talk/OWNS/93/)
by Niels Langeveld (Montanuniversität Leoben) as part of One World Numeration seminar

Abstract
Given the $N$-continued fraction of a number $x$, we construct $N$-continued fraction sequences in the same spirit as Sturmian sequences can be constructed from regular continued fractions. These sequences are infinite words over a two letter alphabet obtained as the limit of a directive sequence of certain substitutions (they are S-adic sequences). By viewing them as a generalisation of Sturmian sequences it is natural to study balancedness. We will see that the sequences we construct are not 1-balanced but C-balanced for $C=N^2$. Furthermore, we construct a dual sequence which is related to the natural extension of the $N$-continued fraction algorithm. This talk is joint work with Lucía Rossi and Jörg Thuswaldner.

David Siukaev (Higher School of Economics)
2022-10-04 12:00 - 12:30 | https://researchseminars.org/talk/OWNS/94/
Title: Exactness and ergodicity of certain Markovian multidimensional fraction algorithms (https://researchseminars.org/talk/OWNS/94/)
by David Siukaev (Higher School of Economics) as part of One World Numeration seminar

Abstract: TBA

Lukas Spiegelhofer (Montanuniversität Leoben)
2022-10-11 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/95/
Title: Primes as sums of Fibonacci numbers (https://researchseminars.org/talk/OWNS/95/)
by Lukas Spiegelhofer (Montanuniversität Leoben) as part of One World Numeration seminar

Abstract
We prove that the Zeckendorf sum-of-digits function of prime numbers, $z(p)$, is uniformly distributed in residue classes.
The main ingredient that made this proof possible is the study of very sparse arithmetic subsequences of $z(n)$. In other words, we will meet the level of distribution.
Our proof of this central result is based on a combination of the "Mauduit−Rivat−van der Corput method" for digital problems and an estimate of a Gowers norm related to $z(n)$.
Our method of proof yields examples of substitutive sequences that are orthogonal to the Möbius function (cf. Sarnak's conjecture).

This is joint work with Michael Drmota and Clemens Müllner (TU Wien).

Alexandra Skripchenko (Higher School of Economics)
2022-10-04 12:30 - 13:00 | https://researchseminars.org/talk/OWNS/96/
Title: Bruin-Troubetzkoy family of interval translation mappings: a new glance (https://researchseminars.org/talk/OWNS/96/)
by Alexandra Skripchenko (Higher School of Economics) as part of One World Numeration seminar

Abstract
In 2002 H. Bruin and S. Troubetzkoy described a special class of interval translation mappings on three intervals. They showed that in this class the typical ITM could be reduced to an interval exchange transformations. They also proved that generic ITM of their class that can not be reduced to IET is uniquely ergodic.

We suggest an alternative proof of the first statement and get a stronger version of the second one. It is a joint work in progress with Mauro Artigiani and Pascal Hubert.

Álvaro Bustos-Gajardo (The Open University)
2022-10-25 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/97/
Title: Quasi-recognizability and continuous eigenvalues of torsion-free S-adic systems (https://researchseminars.org/talk/OWNS/97/)
by Álvaro Bustos-Gajardo (The Open University) as part of One World Numeration seminar

Abstract
We discuss combinatorial and dynamical descriptions of S-adic systems generated by sequences of constant-length morphisms between alphabets of bounded size. For this purpose, we introduce the notion of quasi-recognisability, a strictly weaker version of recognisability but which is indeed enough to reconstruct several classical arguments of the theory of constant-length substitutions in this more general context. Furthermore, we identify a large family of directive sequences, which we call "torsion-free", for which quasi-recognisability is obtained naturally, and can be improved to actual recognisability with relative ease.

Using these notions we give S-adic analogues of the notions of column number and height for substitutions, including dynamical and combinatorial interpretations of each, and give a general characterisation of the maximal equicontinuous factor of the identified family of S-adic shifts, showing as a consequence that in this context all continuous eigenvalues must be rational. As well, we employ the tools developed for a first approach to the measurable case.

This is a joint work with Neil Mañibo and Reem Yassawi.

Wen Wu (South China University of Technology)
2022-11-08 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/98/
Title: From the Thue-Morse sequence to the apwenian sequences (https://researchseminars.org/talk/OWNS/98/)
by Wen Wu (South China University of Technology) as part of One World Numeration seminar

Abstract
In this talk, we will introduce a class of $\pm 1$ sequences, called the apwenian sequences. The Hankel determinants of these $\pm1$ sequences share the same property as the Hankel determinants of the Thue-Morse sequence found by Allouche, Peyrière, Wen and Wen in 1998. In particular, the Hankel determinants of apwenian sequences do not vanish. This allows us to discuss the Diophantine property of the values of their generating functions at $1/b$ where $b\geq 2$ is an integer. Moreover, the number of $\pm 1$ apwenian sequences is given explicitly. Similar questions are also discussed for $0$-$1$ apwenian sequences. This talk is based on joint work with Y.-J. Guo and G.-N. Han.

Faustin Adiceam (Université Paris-Est Créteil)
2022-11-22 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/99/
Title: Badly approximable vectors and Littlewood-type problems (https://researchseminars.org/talk/OWNS/99/)
by Faustin Adiceam (Université Paris-Est Créteil) as part of One World Numeration seminar

Abstract
Badly approximable vectors are fractal sets enjoying rich Diophantine properties. In this respect, they play a crucial role in many problems well beyond Number Theory and Fractal Geometry (e.g., in signal processing, in mathematical physics and in convex geometry).

After outlining some of the latest developments in this very active area of research, we will take an interest in the Littlewood conjecture (c. 1930) and in its variants which all admit a natural formulation in terms of properties satisfied by badly approximable vectors. We will then show how ideas emerging from the mathematical theory of quasicrystals, from numeration systems and from the theory of aperiodic tilings have recently been used to refute the so-called t-adic Littlewood conjecture.

All necessary concepts will be defined in the talk. Joint with Fred Lunnon (Maynooth) and Erez Nesharim (Technion, Haifa).

Yufei Chen (TU Delft)
2022-10-18 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/100/
Title: Matching of orbits of certain $N$-expansions with a finite set of digits (https://researchseminars.org/talk/OWNS/100/)
by Yufei Chen (TU Delft) as part of One World Numeration seminar

Abstract
In this talk we consider a class of continued fraction expansions: the so-called $N$-expansions with a finite digit set, where $N\geq 2$ is an integer. For $N$ fixed they are steered by a parameter $\alpha\in (0,\sqrt{N}-1]$. For $N=2$ an explicit interval $[A,B]$ was determined, such that for all $\alpha\in [A,B]$ the entropy $h(T_{\alpha})$ of the underlying Gauss-map $T_{\alpha}$ is equal. In this paper we show that for all $N\in
$, $N\geq 2$, such plateaux exist. In order to show that the entropy is constant on such plateaux, we obtain the underlying planar natural extension of the maps $T_{\alpha}$, the $T_{\alpha}$-invariant measure, ergodicity, and we show that for any two $\alpha,\alpha'$ from the same plateau, the natural extensions are metrically isomorphic, and the isomorphism is given explicitly. The plateaux are found by a property called matching.

Seul Bee Lee (Institute for Basic Science)
2022-11-15 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/101/
Title: Regularity properties of Brjuno functions associated with by-excess, odd and even continued fractions (https://researchseminars.org/talk/OWNS/101/)
by Seul Bee Lee (Institute for Basic Science) as part of One World Numeration seminar

Abstract
An irrational number is called a Brjuno number if the sum of the series of $\log(q_{n+1})/q_n$ converges, where $q_n$ is the denominator of the $n$-th principal convergent of the regular continued fraction. The importance of Brjuno numbers comes from the study of one variable analytic small divisor problems. In 1988, J.-C. Yoccoz introduced the Brjuno function which characterizes the Brjuno numbers to estimate the size of Siegel disks. In this talk, we introduce Brjuno-type functions associated with by-excess, odd and even continued fractions with a number theoretical motivation. Then we discuss the $L^p$ and the Hölder regularity properties of the difference between the classical Brjuno function and the Brjuno-type functions. This is joint work with Stefano Marmi.

Manuel Hauke (TU Graz)
2022-11-29 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/102/
Title: The asymptotic behaviour of Sudler products (https://researchseminars.org/talk/OWNS/102/)
by Manuel Hauke (TU Graz) as part of One World Numeration seminar

Abstract
Given an irrational number $\alpha$, we study the asymptotic behaviour of the Sudler product defined by $P_N(\alpha) = \prod_{r=1}^N 2 \lvert \sin \pi r \alpha \rvert$, which appears in many different areas of mathematics.
In this talk, we explain the connection between the size of $P_N(\alpha)$ and the Ostrowski expansion of $N$ with respect to $\alpha$.
We show that $\liminf_{N \to \infty} P_N(\alpha) = 0$ and $\limsup_{N \to \infty} P_N(\alpha)/N = \infty$, whenever the sequence of partial quotients in the continued fraction expansion of $\alpha$ exceeds $7$ infinitely often, and show that the value $7$ is optimal.

For Lebesgue-almost every $\alpha$, we can prove more: we show that for every non-decreasing function $\psi: (0,\infty) \to (0,\infty)$ with $\sum_{k=1}^{\infty} \frac{1}{\psi(k)} = \infty$ and
$\liminf_{k \to \infty} \psi(k)/(k \log k)$ sufficiently large, the conditions $\log P_N(\alpha) \leq -\psi(\log N)$, $\log P_N(\alpha) \geq \psi(\log N)$ hold on sets of upper density $1$ respectively $1/2$.

Christoph Bandt (Universität Greifswald)
2022-12-06 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/103/
Title: Automata generated topological spaces and self-affine tilings (https://researchseminars.org/talk/OWNS/103/)
by Christoph Bandt (Universität Greifswald) as part of One World Numeration seminar

Abstract
Numeration assigns symbolic sequences as addresses to points in a space X. There are points which get multiple addresses. It is known that these identifications describe the topology of X and can often be determined by an automaton. Here we define a corresponding class of automata and discuss their properties and interesting examples. Various open questions concern the realization of such automata by iterated functions and the uniqueness of such an implementation. Self-affine tiles form a simple class of examples.

Hiroki Takahasi (Keio University)
2022-12-13 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/104/
Title: Distribution of cycles for one-dimensional random dynamical systems (https://researchseminars.org/talk/OWNS/104/)
by Hiroki Takahasi (Keio University) as part of One World Numeration seminar

Abstract
We consider an independently identically distributed random dynamical system generated by finitely many, non-uniformly expanding Markov interval maps with a finite number of branches.
Assuming a topologically mixing condition and the uniqueness of equilibrium state for the associated skew product map, we establish a samplewise (quenched) almost-sure level-2 weighted equidistribution of "random cycles", with respect to a natural stationary measure as the periods of the cycles tend to infinity. This result implies an analogue of Bowen's theorem on periodic orbits of topologically mixing Axiom A diffeomorphisms.

This talk is based on the preprint arXiv:2108.05522. If time permits, I will mention some future perspectives in this project.

Roswitha Hofer (JKU Linz)
2023-01-10 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/105/
Title: Exact order of discrepancy of normal numbers (https://researchseminars.org/talk/OWNS/105/)
by Roswitha Hofer (JKU Linz) as part of One World Numeration seminar

Abstract
In the talk we discuss some previous results on the discrepancy of normal numbers and consider the still open question of Korobov: What is the best possible order of discrepancy $D_N$ in $N$, a sequence $(\{b^n\alpha\})_{n\geq 0}$, $b\geq 2,\in\mathbb{N}$, can have for some real number $\alpha$? If $\lim_{N\to\infty} D_N=0$ then $\alpha$ in called normal in base $b$.

So far the best upper bounds for $D_N$ for explicitly known normal numbers in base $2$ are of the form $ND_N\ll\log^2 N$. The first example is due to Levin (1999), which was later generalized by Becher and Carton (2019). In this talk we discuss the recent result in joint work with Gerhard Larcher that guarantees $ND_N\gg \log^2 N$ for Levin's binary normal number. So EITHER $ND_N\ll \log^2N$ is the best possible order for $D_N$ in $N$ of a normal number OR there exist another example of a binary normal number with a better growth of $ND_N$ in $N$. The recent result for Levin's normal number might support the conjecture that $ND_N\ll \log^2N$ is the best order for $D_N$ in $N$ a normal number can obtain.

Slade Sanderson (Utrecht University)
2023-01-31 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/106/
Title: Matching for parameterised symmetric golden maps (https://researchseminars.org/talk/OWNS/106/)
by Slade Sanderson (Utrecht University) as part of One World Numeration seminar

Abstract
In 2020, Dajani and Kalle investigated invariant measures and frequencies of digits of signed binary expansions arising from a parameterised family of piecewise linear interval maps of constant slope 2. Central to their study was a property called ‘matching,’ where the orbits of the left and right limits of discontinuity points agree after some finite number of steps. We obtain analogous results for a parameterised family of ‘symmetric golden maps’ of constant slope $\beta$, with $\beta$ the golden mean. Matching is again central to our methods, though the dynamics of the symmetric golden maps are more delicate than the binary case. We characterize the matching phenomenon in our setting, present explicit invariant measures and frequencies of digits of signed $\beta$-expansions, and---time permitting---show further implications for a family of piecewise linear maps which arise as jump transformations of the symmetric golden maps.

Joint with Karma Dajani.

Kiko Kawamura (University of North Texas)
2023-01-24 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/107/
Title: The partial derivative of Okamoto's functions with respect to the parameter (https://researchseminars.org/talk/OWNS/107/)
by Kiko Kawamura (University of North Texas) as part of One World Numeration seminar

Abstract
Okamoto's functions were introduced in 2005 as a one-parameter family of self-affine functions, which are expressed by ternary expansion of $x$ on the interval $[0,1]$. By changing the parameter, one can produce interesting examples: Perkins' nowhere differentiable function, Bourbaki-Katsuura function and Cantor's Devil's staircase function.

In this talk, we consider the partial derivative of Okomoto's functions with respect to the parameter $a$. We place a significant focus on $a = 1/3$ to describe the properties of a nowhere differentiable function $K(x)$ for which the set of points of infinite derivative produces an example of a measure zero set with Hausdorff dimension 1.

This is a joint work with T. Mathis and M.Paizanis (undergraduate students) and N.Dalaklis (graduate student). The talk is very accessible and includes many computer graphics.

Derong Kong (Chongqing University)
2023-03-07 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/108/
Title: Critical values for the beta-transformation with a hole at 0 (https://researchseminars.org/talk/OWNS/108/)
by Derong Kong (Chongqing University) as part of One World Numeration seminar

Abstract
Given $\beta \in (1,2]$, let $T$ be the $\beta$-transformation on the unit circle $[0,1)$. For $t \in [0,1)$ let $K(t)$ be the survivor set consisting of all $x$ whose orbit under $T$ never hits the open interval $(0,t)$. Kalle et al. [ETDS, 2020] proved that the Hausdorff dimension function $\dim K(t)$ is a non-increasing Devil's staircase in $t$. So there exists a critical value such that $\dim K(t)$ is vanishing when $t$ is passing through this critical value. In this paper we will describe this critical value and analyze its interesting properties. Our strategy to find the critical value depends on certain substitutions of Farey words and a renormalization scheme from dynamical systems. This is joint work with Pieter Allaart.

Demi Allen (University of Exeter)
2023-03-21 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/109/
Title: Diophantine Approximation for systems of linear forms - some comments on inhomogeneity, monotonicity, and primitivity (https://researchseminars.org/talk/OWNS/109/)
by Demi Allen (University of Exeter) as part of One World Numeration seminar

Abstract
Diophantine Approximation is a branch of Number Theory in which the central theme is understanding how well real numbers can be approximated by rationals. In the most classical setting, a $\psi$-well-approximable number is one which can be approximated by rationals to a given degree of accuracy specified by an approximating function $\psi$. Khintchine's Theorem provides a beautiful characterisation of the Lebesgue measure of the set of $\psi$-well-approximable numbers and is one of the cornerstone results of Diophantine Approximation. In this talk I will discuss the generalisation of Khintchine's Theorem to the setting of approximation for systems of linear forms. I will focus mainly on the topic of inhomogeneous approximation for systems of linear forms. Time permitting, I may also discuss approximation for systems of linear forms subject to certain primitivity constraints. This talk will be based on joint work with Felipe Ramirez (Wesleyan, US).

Ale Jan Homburg (University of Amsterdam, VU University Amsterdam)
2023-02-07 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/110/
Title: Iterated function systems of linear expanding and contracting maps on the unit interval (https://researchseminars.org/talk/OWNS/110/)
by Ale Jan Homburg (University of Amsterdam, VU University Amsterdam) as part of One World Numeration seminar

Abstract
We analyze the two-point motions of iterated function systems on the unit interval generated by expanding and contracting affine maps, where the expansion and contraction rates are determined by a pair $(M,N)$ of integers.

This dynamics depends on the Lyapunov exponent.

For a negative Lyapunov exponent we establish synchronization, meaning convergence of orbits with different initial points. For a vanishing Lyapunov exponent we establish intermittency, where orbits are close for a set of iterates of full density, but are intermittently apart. For a positive Lyapunov exponent we show the existence of an absolutely continuous stationary measure for the two-point dynamics and discuss its consequences.

For nonnegative Lyapunov exponent and pairs $(M,N)$ that are multiplicatively dependent integers, we provide explicit expressions for absolutely continuous stationary measures of the two-point dynamics. These stationary measures are infinite $\sigma$-finite measures in the case of zero Lyapunov exponent.

This is joint work with Charlene Kalle.

Yining Hu (Huazhong University of Science and Technology)
2023-02-14 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/111/
Title: Algebraic automatic continued fractions in characteristic 2 (https://researchseminars.org/talk/OWNS/111/)
by Yining Hu (Huazhong University of Science and Technology) as part of One World Numeration seminar

Abstract
We present two families of automatic sequences that define algebraic continued fractions in characteristic $2$. The period-doubling sequence belongs to the first family $\mathcal{P}$; and its sum modulo $2$, the Thue-Morse sequence, belongs to the second family $\mathcal{G}$. The family $\mathcal{G}$ contains all the iterated sums of sequences from the $\mathcal{P}$ and more.

Roland Zweimüller (University of Vienna)
2023-03-28 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/112/
Title: Variations on a theme of Doeblin (https://researchseminars.org/talk/OWNS/112/)
by Roland Zweimüller (University of Vienna) as part of One World Numeration seminar

Abstract
Starting from Doeblin's observation on the Poissonian nature of occurrences of large digits in typical continued fraction expansions, I will outline some recent work on rare events in measure preserving systems (including spatiotemporal and local limit theorems) which, in particular, allows us to refine Doeblin's statement in several ways.

(Part of this is joint work with Max Auer.)

Anton Lukyanenko (George Mason University)
2023-04-18 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/113/
Title: Serendipitous decompositions of higher-dimensional continued fractions (https://researchseminars.org/talk/OWNS/113/)
by Anton Lukyanenko (George Mason University) as part of One World Numeration seminar

Abstract
Complex continued fractions (CFs) represent a complex number using a descending fraction with Gaussian integer coefficients. The associated dynamical system is exact (Nakada 1981) with a piecewise-analytic invariant measure (Hensley 2006). Certain higher-dimensional CFs, including CFs over quaternions, octonions, as well as the non-commutative Heisenberg group can be understood in a unified way using the Iwasawa CF framework (L-Vandehey 2022). Under some natural and robust assumptions, ergodicity of the associated systems can then be derived from a connection to hyperbolic geodesic flow, but stronger mixing results and information about the invariant measure remain elusive. Here, we study Iwasawa CFs under a more delicate serendipity assumption that yields the finite range condition, allowing us to extend the Nakada-Hensley results to certain Iwasawa CFs over the quaternions, octonions, and in $\mathbb{R}^3$.

This is joint work with Joseph Vandehey.

Ronnie Pavlov (University of Denver)
2023-04-25 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/114/
Title: Subshifts of very low complexity (https://researchseminars.org/talk/OWNS/114/)
by Ronnie Pavlov (University of Denver) as part of One World Numeration seminar

Abstract
The word complexity function $p(n)$ of a subshift $X$ measures the number of $n$-letter words appearing in sequences in $X$, and $X$ is said to have linear complexity if $p(n)/n$ is bounded. It's been known since work of Ferenczi that subshifts X with linear word complexity function (i.e. $\limsup p(n)/n$ finite) have highly constrained/structured behavior. I'll discuss recent work with Darren Creutz, where we show that if $\limsup p(n)/n < 4/3$, then the subshift $X$ must in fact have measurably discrete spectrum, i.e. it is isomorphic to a compact abelian group rotation. Our proof uses a substitutive/S-adic decomposition for such shifts, and I'll touch on connections to the so-called S-adic Pisot conjecture.

Craig S. Kaplan (University of Waterloo)
2023-05-09 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/115/
Title: An aperiodic monotile (https://researchseminars.org/talk/OWNS/115/)
by Craig S. Kaplan (University of Waterloo) as part of One World Numeration seminar

Abstract
A set of shapes is called aperiodic if the shapes admit tilings of the plane, but none that have translational symmetry. A longstanding open problem asks whether a set consisting of a single shape could be aperiodic; such a shape is known as an aperiodic monotile or sometimes an "einstein". The recently discovered "hat" monotile settles this problem in two dimensions. In this talk I provide necessary background on aperiodicity and related topics in tiling theory, review the history of the search for for an aperiodic monotile, and then discuss the hat and its mathematical properties.

Mark Pollicott (University of Warwick)
2023-09-05 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/116/
Title: Complex Dimensions and Fractal Strings (https://researchseminars.org/talk/OWNS/116/)
by Mark Pollicott (University of Warwick) as part of One World Numeration seminar

Abstract
Some years ago M.Lapidus introduced the notion of complex dimensions for a Cantor set in the real line. These occur as poles of the complex Dirichlet series formed from the lengths of the bounded intervals (the "fractal strings") in the complement of the Cantor set. We will explore further these ideas when the Cantor set is the attractor of an iterated function scheme (concentrating on those whose contractions are a finite set of inverse branches of the usual Gauss map).

James Worrell (University of Oxford)
2023-09-19 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/117/
Title: Transcendence of Sturmian Numbers over an Algebraic Base (https://researchseminars.org/talk/OWNS/117/)
by James Worrell (University of Oxford) as part of One World Numeration seminar

Abstract
Ferenczi and Mauduit showed in 1997 that a number represented over an integer base by a Sturmian sequence of digits is transcendental. In this talk we generalise this result to hold for all algebraic number base b of absolute value strictly greater than one. More generally, for a given base b and given irrational number θ, we prove rational linear independence of the set comprising 1 together with all numbers of the above form whose associated digit sequences have slope θ.

We give an application of our main result to the theory of dynamical systems. We show that for a Cantor set C arising as the set of limit points of a contracted rotation f on the unit interval, where f is assumed to have an algebraic slope, all elements of C except its endpoints 0 and 1 are transcendental.

This is joint work with Florian Luca and Joel Ouaknine.

Manfred Madritsch (Université de Lorraine)
2023-10-03 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/118/
Title: Construction of absolutely normal numbers (https://researchseminars.org/talk/OWNS/118/)
by Manfred Madritsch (Université de Lorraine) as part of One World Numeration seminar

Abstract
Let $b\geq2$ be a positive integer. Then every real number $x\in[0,1]$ admits a
$b$-adic representation with digits $a_k$. We call the real $x$ simply
normal to base $b$ if every digit $d\in\{0,1,\dots,b-1\}$ occurs with the same
frequency in the $b$-ary representation. Furthermore we call $x$ normal to
base $b$, if it is simply normal with respect to $b$, $b^2$, $b^3$, etc.
Finally we call $x$ absolutely normal if it is normal with respect to all
bases $b\geq2$.

In the present talk we want to generalize this notion to normality in measure
preserving systems like $\beta$-expansions and continued fraction expansions.
Then we show constructions of numbers that are (absolutely) normal with respect
to several different expansions.

Fumichika Takamizo (Osaka Metropolitan University)
2023-10-17 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/119/
Title: Finite $\beta$-expansion of natural numbers (https://researchseminars.org/talk/OWNS/119/)
by Fumichika Takamizo (Osaka Metropolitan University) as part of One World Numeration seminar

Abstract
If $\beta$ is an integer, then each $x \in \mathbb{Z}[1/\beta] \cap [0,\infty)$ has finite expansion in base $\beta$. As a generalization of this property for $\beta>1$, the condition (F$_{1}$) that each $x \in \mathbb{N}$ has finite $\beta$-expansion was proposed by Frougny and Solomyak.
In this talk, we give a sufficient condition for (F$_{1}$). Moreover we also find $\beta$ with property (F$_{1}$) which does not have positive finiteness property.

Stefano Marmi (Scuola Normale Superiore)
2023-10-31 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/120/
Title: Complexified continued fractions and complex Brjuno and Wilton functions (https://researchseminars.org/talk/OWNS/120/)
by Stefano Marmi (Scuola Normale Superiore) as part of One World Numeration seminar

Abstract
We study functions related to the classical Brjuno function, namely k-Brjuno functions and the Wilton function. Both appear in the study of boundary regularity properties of (quasi) modular forms and their integrals. We then complexify the functional equations which they fulfill and we construct analytic extensions of the k-Brjuno and Wilton functions to the upper half-plane. We study their boundary behaviour using an extension of the continued fraction algorithm to the complex plane. We also prove that the harmonic conjugate of the real k-Brjuno function is continuous at all irrational numbers and has a decreasing jump of π/qk at rational points p/q. This is based on joint work with S. B. Lee, I. Petrykiewicz and T. I. Schindler, the paper is available (open source) at this link: https://link.springer.com/article/10.1007/s00010-023-00967-w

Jana Lepšová (Czech Technical University in Prague, Université de Bordeaux)
2023-11-14 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/121/
Title: Dumont-Thomas numeration systems for ℤ (https://researchseminars.org/talk/OWNS/121/)
by Jana Lepšová (Czech Technical University in Prague, Université de Bordeaux) as part of One World Numeration seminar

Abstract
We extend the well-known Dumont-Thomas numeration system to ℤ by considering two-sided periodic points of a substitution, thus allowing us to represent any integer in ℤ by a finite word (starting with 0 when nonnegative and with 1 when negative). We show that an automaton returns the letter at position $n \in ℤ$ of the periodic point when fed with the representation of $n$. The numeration system naturally extends to $ℤ^d$. We give an equivalent characterization of the numeration system in terms of a total order on a regular language. Lastly, using particular periodic points, we recover the well-known two's complement numeration system and the Fibonacci analogue of the two's complement numeration system.

Claudio Bonanno (Università di Pisa)
2023-11-28 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/122/
Title: Asymptotic behaviour of the sums of the digits for continued fraction algorithms (https://researchseminars.org/talk/OWNS/122/)
by Claudio Bonanno (Università di Pisa) as part of One World Numeration seminar

Abstract
In this talk I will discuss applications of methods of ergodic theory to obtain pointwise asymptotic behaviour for the sum of the digits of some non-regular continued fraction algorithms. The idea is to study the behaviour of trimmed Birkhoff sums for infinite-measure preserving dynamical systems. The talk is based on joint work with Tanja I. Schindler.

Yasushi Nagai (Shinshu University)
2023-12-12 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/123/
Title: Overlap algorithm for general S-adic tilings (https://researchseminars.org/talk/OWNS/123/)
by Yasushi Nagai (Shinshu University) as part of One World Numeration seminar

Abstract
We investigate the question of when a tiling has pure point spectrum, for the class of $S$-adic tilings, which includes all self-affine tilings. The overlap algorithm by Solomyak is a powerful tool to study this problem for the class of self-affine tilings. We generalize this algorithm for general $S$-adic tilings, and apply it to a class of block $S$-adic tilings to show almost all of them have pure point spectra. This is a joint work with Jörg Thuswaldner.

Karma Dajani (Universiteit Utrecht)
2024-01-16 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/124/
Title: Alternating N-continued fraction expansions (https://researchseminars.org/talk/OWNS/124/)
by Karma Dajani (Universiteit Utrecht) as part of One World Numeration seminar

Abstract
We introduce a family of maps generating continued fractions where the digit 1 in the numerator is replaced cyclically by some given non-negative integers $(N_1, \dots, N_m)$. We prove the convergence of the given algorithm, and study the underlying dynamical system generating such expansions. We prove the existence of a unique absolutely continuous invariant ergodic measure. In special cases, we are able to build the natural extension and give an explicit expression of the invariant measure. For these cases, we formulate a Doeblin-Lenstra type theorem. For other cases we have a more implicit expression that we conjecture gives the invariant density. This conjecture is supported by simulations. For the simulations we use a method that gives us a smooth approximation in every iteration. This is joint work with Niels Langeveld.

Cathy Swaenepoel (Université Paris Cité)
2024-01-30 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/125/
Title: Reversible primes (https://researchseminars.org/talk/OWNS/125/)
by Cathy Swaenepoel (Université Paris Cité) as part of One World Numeration seminar

Abstract
The properties of the digits of prime numbers and various other
sequences of integers have attracted great interest in recent years.
For any positive integer $k$, we denote by $\overleftarrow{k}$ the
reverse of $k$ in base 2, defined by $\overleftarrow{k} = \sum_{j=0}^{n-1} \varepsilon_j\,2^{n-1-j}$ where $k = \sum_{j=0}^{n-1} \varepsilon_{j} \,2^j$ with $\varepsilon_j \in \{0,1\}$, $j\in\{0, \ldots, n-1\}$, $ \varepsilon_{n-1} = 1$. A natural question is to estimate the number
of primes $p\in \left[2^{n-1},2^n\right)$ such that
$\overleftarrow{p}$ is prime. We will present a result which provides
an upper bound of the expected order of magnitude. Our method is based
on a sieve argument and also allows us to obtain a strong lower bound
for the number of integers $k$ such that $k$ and $\overleftarrow{k}$
have at most 8 prime factors (counted with multiplicity). We will also
present an asymptotic formula for the number of integers
$k\in \left[2^{n-1},2^n\right)$ such that $k$ and $\overleftarrow{k}$
are squarefree.

This is a joint work with Cécile Dartyge, Bruno Martin,
Joël Rivat and Igor Shparlinski.

Bartosz Sobolewski (Jagiellonian University in Kraków and Montanuniversität Leoben)
2024-02-13 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/126/
Title: Block occurrences in the binary expansion of n and n+t (https://researchseminars.org/talk/OWNS/126/)
by Bartosz Sobolewski (Jagiellonian University in Kraków and Montanuniversität Leoben) as part of One World Numeration seminar

Abstract
Let $s(n)$ denote the sum of binary digits of a nonnegative integer $n$. In 2012 Cusick asked whether for every nonnegative integer $t$ the set of $n$ satisfying $s(n+t) \geq s(n)$ has natural density strictly greater than $1/2$. So far it is known that the answer is affirmative for almost all $t$ (in the sense of density) and $s(n+t) - s(n)$ has approximately Gaussian distribution. During the talk we consider an analogue of this problem concerning the function $r(n)$, which counts the occurrences of the block $11$ in the binary expansion of $n$. In particular, we prove that the distribution of $r(n+t)-r(n)$ is approximately Gaussian as well. We also discuss a generalization to an arbitrary block of binary digits. This is a joint work with Lukas Spiegelhofer.

Joël Ouaknine (Max Planck Institute for Software Systems)
2024-03-12 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/127/
Title: The Skolem Landscape (https://researchseminars.org/talk/OWNS/127/)
by Joël Ouaknine (Max Planck Institute for Software Systems) as part of One World Numeration seminar

Abstract
The Skolem Problem asks how to determine algorithmically whether a given linear recurrence sequence (such as the Fibonacci numbers) has a zero. It is a central question in dynamical systems and number theory, and has many connections to other branches of mathematics and computer science. Unfortunately, its decidability has been open for nearly a century! In this talk, I will present a survey of what is known on the Skolem Problem and related questions, including recent and ongoing developments.

Nikita Shulga (La Trobe University)
2024-03-26 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/128/
Title: Radical bound for Zaremba’s conjecture (https://researchseminars.org/talk/OWNS/128/)
by Nikita Shulga (La Trobe University) as part of One World Numeration seminar

Abstract
Zaremba's conjecture states that for each positive integer $q$, there exists a coprime integer $a$, smaller than $q$, such that partial quotients in the continued fraction expansion of $a/q$ are bounded by some absolute constant. Despite major breakthroughs in the recent years, the conjecture is still open. In this talk I will discuss a new result towards Zaremba's conjecture, proving that for each denominator, one can find a numerator, such that partial quotients are bounded by the radical of the denominator, i.e. the product of distinct prime factors. This generalizes the result by Niederreiter and improves upon some results of Moshchevitin-Murphy-Shkredov.

Shunsuke Usuki (Kyoto University)
2024-04-23 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/130/
Title: On a lower bound of the number of integers in Littlewood's conjecture (https://researchseminars.org/talk/OWNS/130/)
by Shunsuke Usuki (Kyoto University) as part of One World Numeration seminar

Abstract
Littlewood's conjecture is a famous and long-standing open problem which states that, for every $(\alpha,\beta) \in \mathbb{R}^2$, $n\|n\alpha\|\|n\beta\|$ can be arbitrarily small for some integer $n$.
This problem is closely related to the action of diagonal matrices on $\mathrm{SL}(3,\mathbb{R})/\mathrm{SL}(3,\mathbb{Z})$, and a groundbreaking result was shown by Einsiedler, Katok and Lindenstrauss from the measure rigidity for this action, saying that Littlewood's conjecture is true except on a set of Hausdorff dimension zero.
In this talk, I will explain about a new quantitative result on Littlewood's conjecture which gives, for every $(\alpha,\beta) \in \mathbb{R}^2$ except on sets of small Hausdorff dimension, an estimate of the number of integers $n$ which make $n\|n\alpha\|\|n\beta\|$ small. The keys for the proof are the measure rigidity and further studies on behavior of empirical measures for the diagonal action.

Tom Kempton (University of Manchester)
2024-05-07 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/131/
Title: The Dynamics of the Fibonacci Partition Function (https://researchseminars.org/talk/OWNS/131/)
by Tom Kempton (University of Manchester) as part of One World Numeration seminar

Abstract
The Fibonacci partition function $R(n)$ counts the number of ways of representing a natural number $n$ as the sum of distinct Fibonacci numbers. For example, $R(6)=2$ since $6=5+1$ and $6=3+2+1$. An explicit formula for $R(n)$ was recently given by Chow and Slattery. In this talk we express $R(n)$ in terms of ergodic sums over an irrational rotation, which allows us to prove lots of statements about the local structure of $R(n)$.

Gaétan Guillot (Université Paris-Saclay)
2024-05-21 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/132/
Title: Approximation of linear subspaces by rational linear subspaces (https://researchseminars.org/talk/OWNS/132/)
by Gaétan Guillot (Université Paris-Saclay) as part of One World Numeration seminar

Abstract
We elaborate on a problem raised by Schmidt in 1967: rational approximation of linear subspaces of $\mathbb{R}^n$. In order to study the quality approximation of irrational numbers by rational ones, one can introduce the exponent of irrationality of a number. We can then generalize this notion in the framework of vector subspaces for the approximation of a subspace by so-called rational subspaces.

After briefly introducing the tools for constructing this generalization, I will present the different possible studies of this object. Finally I will explain how we can construct spaces with prescribed exponents.

Noy Soffer Aranov (Technion)
2024-06-18 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/133/
Title: Escape of Mass of the Thue Morse Sequence (https://researchseminars.org/talk/OWNS/133/)
by Noy Soffer Aranov (Technion) as part of One World Numeration seminar

Abstract
One way to study the distribution of quadratic number fields is through the evolution of continued fraction expansions. In the function field setting, it was shown by de Mathan and Teullie that given a quadratic irrational $\Theta$, the degrees of the periodic part of the continued fraction of $t^n\Theta$ are unbounded. Paulin and Shapira improved this by proving that quadratic irrationals exhibit partial escape of mass. Moreover, they conjectured that they must exhibit full escape of mass. We show that the Thue Morse sequence is a counterexample to their conjecture. In this talk we shall discuss the technique of proof as well as the connection between escape of mass in continued fractions, Hecke trees, and number walls. This is part of ongoing work joint with Erez Nesharim.

Simon Kristensen (Aarhus Universitet)
2024-09-17 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/134/
Title: On the distribution of sequences of the form $(q_n y)$ (https://researchseminars.org/talk/OWNS/134/)
by Simon Kristensen (Aarhus Universitet) as part of One World Numeration seminar

Abstract
The distribution of sequences of the form $(q_n y)$ with $(q_n)$ a sequence of integers and $y$ a real number have attracted quite a bit of attention, for instance due to their relation to inhomogeneous Littlewood type problems. In this talk, we will provide some results on the Lebesgue measure and Hausdorff dimension on the set of points in the unit interval approximated to a certain rate by points from such a sequence. A feature of our approach is that we obtain estimates even in the case when the sequence $(q_n)$ grows rather slowly. This is joint work with Tomas Persson.

Dong Han Kim (Dongguk University)
2024-10-01 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/135/
Title: Uniform Diophantine approximation on the Hecke group $H_4$ (https://researchseminars.org/talk/OWNS/135/)
by Dong Han Kim (Dongguk University) as part of One World Numeration seminar

Abstract
Dirichlet's uniform approximation theorem is a fundamental result in Diophantine approximation that gives an optimal rate of approximation.
We study uniform Diophantine approximation properties on the Hecke group $H_4$ in terms of the Rosen continued fractions.
For a given real number $\alpha$, the best approximations are convergents of the Rosen continued fraction and the dual Rosen continued fraction of $\alpha$.
We give analogous theorems of Dirichlet uniform approximation and the Legendre theorem with optimal constants.
This is joint work with Ayreena Bakhtawar and Seul Bee Lee.

Florian Luca (Stellenbosch University)
2024-11-12 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/136/
Title: On a question of Douglass and Ono (https://researchseminars.org/talk/OWNS/136/)
by Florian Luca (Stellenbosch University) as part of One World Numeration seminar

Abstract
It is known that the partition function $p(n)$ obeys Benford's law in any integer base $b\ge 2$. A similar result was obtained by Douglass and Ono for the plane partition function $\text{PL}(n)$ in a recent paper. In their paper, Douglass and Ono asked for an explicit version of this result. In particular, given an integer base $b\ge 2$ and string $f$ of digits in base $b$ they asked for an explicit value $N(b,f)$ such that there exists $n\le N(b,f)$ with the property that $\text{PL}(n)$ starts with the string $f$ when written in base $b$. In my talk, I will present an explicit value for $N(b,f)$ both for the partition function $p(n)$ as well as for the plane partition function $\text{PL}(n)$.

Haojie Ren (Technion)
2024-11-26 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/137/
Title: The dimension of Bernoulli convolutions in $\mathbb{R}^d$ (https://researchseminars.org/talk/OWNS/137/)
by Haojie Ren (Technion) as part of One World Numeration seminar

Abstract
For $(\lambda_{1},\dots,\lambda_{d})=\lambda\in(0,1)^{d}$ with $\lambda_{1}>\cdots>\lambda_{d}$,
denote by $\mu_{\lambda}$ the Bernoulli convolution associated to
$\lambda$. That is, $\mu_{\lambda}$ is the distribution of the random
vector $\sum_{n\ge0}\pm\left(\lambda_{1}^{n},\dots,\lambda_{d}^{n}\right)$,
where the $\pm$ signs are chosen independently and with equal weight.
Assuming for each $1\le j\le d$ that $\lambda_{j}$ is not a root
of a polynomial with coefficients $\pm1,0$, we prove that the dimension
of $\mu_{\lambda}$ equals $\min\{ \dim_{L}\mu_{\lambda},d\} $,
where $\dim_{L}\mu_{\lambda}$ is the Lyapunov dimension. This is a joint work with Ariel Rapaport.

Steven Robertson (University of Manchester)
2024-10-15 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/138/
Title: Low Discrepancy Digital Hybrid Sequences and the $t$-adic Littlewood Conjecture (https://researchseminars.org/talk/OWNS/138/)
by Steven Robertson (University of Manchester) as part of One World Numeration seminar

Abstract
The discrepancy of a sequence measures how quickly it approaches a uniform distribution. Given a natural number $d$, any collection of one-dimensional so-called low discrepancy sequences $\{S_i : 1 \le i \le d\}$ can be concatenated to create a $d$-dimensional hybrid sequence $(S_1, . . . , S_d)$. Since their introduction by Spanier in 1995, many connections between the discrepancy of a hybrid sequence and the discrepancy of its component sequences have been discovered. However, a proof that a hybrid sequence is capable of being low discrepancy has remained elusive. In this talk, an explicit connection between Diophantine approximation over function fields and two dimensional low discrepancy hybrid sequences is provided.

Specifically, it is shown that any counterexample to the so-called $t$-adic Littlewood Conjecture ($t$-LC) can be used to create a low discrepancy digital Kronecker-Van der Corput sequence. Such counterexamples to $t$-LC are known explicitly over a number of finite fields by, on the one hand, Adiceam, Nesharim and Lunnon, and on the other, by Garrett and the Robertson. All necessary concepts will be defined in the talk.

Victor Shirandami (University of Manchester)
2024-10-29 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/139/
Title: Probabilistic Effectivity in the Subspace Theorem and the Distribution of Algebraic Projective Points (https://researchseminars.org/talk/OWNS/139/)
by Victor Shirandami (University of Manchester) as part of One World Numeration seminar

Abstract
The celebrated Roth’s theorem in Diophantine Approximation determines the degree to which an algebraic number may be approximated by rationals. A corollary of this theorem yields a transcendence criterion for real numbers based off of their decimal expansion. This theorem, and its broad generalisation due to Schmidt, famously suffers from ineffectivity. This motivates one to address this issue in the probabilistic context, whereby one makes progress in the direction of effectivity in an appropriately defined probabilistic regime. From this analysis is derived an analogue of Khintchine's theorem for algebraic numbers, answering a question of Beresnevich, Bernick, and Dodson on a density version of Waldschmidt’s conjecture.

Ali Messaoudi (Universidade Estadual Paulista)
2024-12-10 12:30 - 13:15 | https://researchseminars.org/talk/OWNS/140/
Title: Adding machine, automata and Julia sets (https://researchseminars.org/talk/OWNS/140/)
by Ali Messaoudi (Universidade Estadual Paulista) as part of One World Numeration seminar

Abstract
A stochastic adding machine (defined by P.R. Killeen and T.J. Taylor) is a Markov chain whose states are natural integers, which models the process of adding the number 1 but where there is a probability of failure in which a carry is not performed when necessary. In this lecture, we will talk about dynamical, spectral and probabilistic properties of extensions for the stochastic adding machine and their connections with other topics as Julia sets, Automata and Dynamical Systems on Banach spaces.

Jean-Paul Allouche (CNRS, Sorbonne Université)
2025-01-07 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/141/
Title: Kolam, Ethnomathematics, and Morphisms (https://researchseminars.org/talk/OWNS/141/)
by Jean-Paul Allouche (CNRS, Sorbonne Université) as part of One World Numeration seminar

Abstract
Kolam is a form of traditional decorative art in India, that is drawn by using rice flour, white stone powder, chalk or chalk powder. It is often practised by women in front of their house entrance. One particular kolam uses a 4x4 grid. It can also be found in the Vanuatu Islands, in Africa, etc. We show that a natural generalization on grids of size 8x8, 16x16, etc. is linked to... the Thue-Morse sequence. Further we unveil two (twin) morphisms that generate this family of kolam, and show that they appear in unrelated and somewhat unexpected fields. Time permitting we will allude to a vast, relatively new, field: ethnomathematic(s).

Thomas Garrity (Williams College)
2025-01-21 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/142/
Title: Multi-dimensional continued fractions and integer partitions: Using the Natural Extension to create a tree structure on partitions (https://researchseminars.org/talk/OWNS/142/)
by Thomas Garrity (Williams College) as part of One World Numeration seminar

Abstract: TBA

Giulia Salvatori (Politecnico di Torino)
2025-02-04 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/143/
Title: Continued Fractions, Quadratic Forms, and Regulator Computation for Integer Factorization (https://researchseminars.org/talk/OWNS/143/)
by Giulia Salvatori (Politecnico di Torino) as part of One World Numeration seminar

Abstract
In the realm of integer factorization, certain methods, such as CFRAC, leverage the properties of continued fractions, while others, like SQUFOF, combine these properties with the tools provided by quadratic forms. Recently, Michele Elia revisited the fundamental concepts of SQUFOF, including reduced quadratic forms, distance between quadratic forms, and Gauss composition, offering a new perspective for designing factorization methods.

In this seminar, we present our algorithm, which is a refinement of Elia's method, along with a precise analysis of its computational cost.
Our algorithm is polynomial-time, provided knowledge of a (not too large) multiple of the regulator of $\mathbb{Q}(\sqrt{N})$.
The computation of the regulator governs the total computational cost, which is subexponential, and in particular $O(\exp(\frac{3}{\sqrt{8}}\sqrt{\ln N \ln \ln N}))$.
This makes our method more efficient than CFRAC and SQUFOF, though less efficient than the General Number Field Sieve.

We identify a broad family of integers to which our method is applicable including certain classes of RSA moduli.
Finally, we introduce some promising avenues for refining our method. These span several areas, ranging from Algebraic Number Theory, particularly for estimating the size of the regulator of $\mathbb{Q}(\sqrt{N})$, to Analytic Number Theory, particularly for computing a specific class of $L$-functions.

Joint work with Nadir Murru.

Neil MacVicar (Queen's University)
2025-02-18 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/144/
Title: Intersecting Cantor sets generated by Complex Radix Expansions (https://researchseminars.org/talk/OWNS/144/)
by Neil MacVicar (Queen's University) as part of One World Numeration seminar

Abstract
Consider the classical middle third Cantor set. This is a self-similar set containing all the numbers in the unit interval which have a ternary expansion that avoids the digit 1. We can ask when the intersection of the Cantor set with a translate of itself is also self-similar. Sufficient and necessary conditions were given by Deng, He, and Wen in 2008. This question has also been generalized to classes of subsets of the unit interval. I plan to discuss how existing ideas can be used to address the question for certain self-similar sets with dimension greater than one. These ideas will be illustrated using a class of self-similar sets in the plane that can be realized as radix expansions in base $-n+i$ where $n$ is a positive integer. I will also discuss a property of the fractal dimensions of these kinds of intersections.

Valentin Ovsienko (CNRS, Université de Reims-Champagne-Ardenne)
2025-03-18 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/145/
Title: From Catalan numbers to integrable dynamics: continued fractions and Hankel determinants for q-numbers (https://researchseminars.org/talk/OWNS/145/)
by Valentin Ovsienko (CNRS, Université de Reims-Champagne-Ardenne) as part of One World Numeration seminar

Abstract
The classical Catalan and Motzkin numbers have remarkable continued fraction expansions, the corresponding sequences of Hankel determinants consist of -1, 0 and 1 only. We find an infinite family of power series corresponding to q-deformed real numbers that have very similar properties. Moreover, their sequences of Hankel determinants turn out to satisfy Somos and Gale-Robinson recurrences. (Partially based on a joint work with Emmanuel Pedon.)

Meng Wu (Oulun yliopisto)
2025-04-01 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/146/
Title: On normal numbers in fractals (https://researchseminars.org/talk/OWNS/146/)
by Meng Wu (Oulun yliopisto) as part of One World Numeration seminar

Abstract
Let $K$ be the ternary Cantor set, and let $\mu$ be the Cantor–Lebesgue measure on $K$. It is well known that every point in $K$ is not 3-normal. However, if we take any natural number $p \ge 2$ that is not a power of 3, then $\mu$-almost every point in $K$ is $p$-normal. This classical result is due to Cassels and W. Schmidt.

Another way to obtain normal numbers from K is by rescaling and translating $K$, then examining the transformed set. A recent nice result by Dayan, Ganguly, and Barak Weiss shows that for any irrational number $t$, for $\mu$-almost all $x \in K$, the product $tx$ is 3-normal.

In this talk, we will discuss these results and their generalizations, including replacing $p$ with an arbitrary beta number and considering more general times-3 invariant measures instead of the Cantor–Lebesgue measure.

James Cumberbatch (Purdue University)
2025-04-15 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/147/
Title: Smooth numbers with restricted digits (https://researchseminars.org/talk/OWNS/147/)
by James Cumberbatch (Purdue University) as part of One World Numeration seminar

Abstract
Integers obeying a digital restriction, such as having no 7s in their base 10 representation, are a discrete analog of the Cantor set and have been a recent topic of interest in analytic number theory. Smooth integers, which are integers having only small prime factors, are an important class of integers with applications to many different areas of math. In this talk, we find an asymptotic on the intersection between the two.

Yan Huang (Chongqing University)
2025-04-29 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/148/
Title: The Coincidence of Rényi–Parry Measures for β-Transformation (https://researchseminars.org/talk/OWNS/148/)
by Yan Huang (Chongqing University) as part of One World Numeration seminar

Abstract
We present a complete characterization of two non-integers with the same Rényi-Parry measure.
We prove that for two non-integers $\beta_1 ,\beta_2 >1$, the Rényi-Parry measures coincide if and only if $\beta_1$ is the root of equation $x^2-qx-p=0$, where $p,q\in\mathbb{N}$ with $p\leq q$, and $\beta_2 = \beta_1 + 1$, which confirms a conjecture of Bertrand-Mathis in [A. Bertrand-Mathis, Acta Math. Hungar. 78, no. 1-2 (1998):71–78].

Artem Dudko (IM PAN)
2025-05-13 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/149/
Title: On attractors of Fibonacci maps (https://researchseminars.org/talk/OWNS/149/)
by Artem Dudko (IM PAN) as part of One World Numeration seminar

Abstract
In 1990s Bruin, Keller, Nowicki, and van Strien showed that smooth unimodal maps with Fibonacci combinatorics and sufficiently high degree of a critical point have a wild attractor, i.e. their metric and topological attractors do not coincide. However, until now there were no reasonable estimates on the degree of the critical point needed.

In the talk I will present an approach for studying attractors of maps, which are periodic points of a renormalization. Using this approach and rigorous computer estimates, we show that the Fibonacci map of degree $d=3.8$ does not have a wild attractor, but that for degree $d=5.1$ the wild attractor exists. The talk is based on a joint work with Denis Gaidashev.

Savinien Kreczman (Université de Liège)
2025-05-27 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/150/
Title: Positionality for Dumont-Thomas numeration systems (https://researchseminars.org/talk/OWNS/150/)
by Savinien Kreczman (Université de Liège) as part of One World Numeration seminar

Abstract
Dumont-Thomas numeration systems are a subclass of abstract numeration systems where the factorisation of the fixed point of a substitution is used to represent numbers. A positional numeration system is one where a weight can be assigned to each position so that the evaluation map is an inner product with the weights. For general abstract numeration systems, deciding positionality is an open problem. In this talk, we define an extension of Dumont-Thomas numeration systems to all integers. We then offer a criterion for deciding the positionality of such a system. If time permits, we show a link to Bertrand numeration systems, another familiar class of numeration systems.

Joint work with Sébastien Labbé and Manon Stipulanti.

Yuta Suzuki (Rikkyo University)
2025-06-10 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/151/
Title: Telhcirid's theorem on arithmetic progressions (https://researchseminars.org/talk/OWNS/151/)
by Yuta Suzuki (Rikkyo University) as part of One World Numeration seminar

Abstract
The classical Dirichlet theorem on arithmetic progressions states that there are infinitely many primes in a given arithmetic progression with a trivial necessary condition. In this talk, we prove a "reversed" version of this theorem, which may be called Telhcirid's theorem on arithmetic progressions, i.e., we prove that there are infinitely many primes whose reverse of radix representation is in a given arithmetic progression except in some degenerate cases. This is a joint work with Gautami Bhowmik (University of Lille).

Vilmos Komornik (Université de Strasbourg)
2025-09-23 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/152/
Title: Topology of univoque sets in double-base expansions (https://researchseminars.org/talk/OWNS/152/)
by Vilmos Komornik (Université de Strasbourg) as part of One World Numeration seminar

Abstract
This is a joint work with Yuru Zou and YiChang Li. Given two real numbers $q_0,q_1>1$ satisfying $q_0+q_1\geq q_0q_1$ and two real numbers $d_0\ne d_1$, by a double-base expansion of a real number $x$ we mean a sequence $(i_k)\in \{0,1\}^{\infty}$ such that
\[
x=\sum_{k=1}^{\infty}\frac{d_{i_k}}{q_{{i_1}}q_{{i_2}}\cdots q_{{i_k}}}.
\]
We denote by $\mathcal{U}_{{q_0,q_1}}$ the set of numbers $x$ having a unique expansion.
The topological properties of $\mathcal{U}_{{q_0,q_1}}$ have been investigated in the equal-base case $q_0=q_1$ for a long time.
We extend this research to the case $q_0\neq q_1$.
While many results remain valid, a great number of new phenomena appear due to the increased complexity of double-base expansions.

Anna Chiara Lai (Sapienza Università di Roma)
2025-10-07 11:00 - 12:00 | https://researchseminars.org/talk/OWNS/153/
Title: Optimal expansions of Kakeya sequences (https://researchseminars.org/talk/OWNS/153/)
by Anna Chiara Lai (Sapienza Università di Roma) as part of One World Numeration seminar

Abstract
Expansions of Kakeya sequences generalize the expansions in non-integer bases and they display analogous redundancy phenomena. In this talk, we present a characterization of optimal expansions of Kakeya sequences, and we provide conditions for the existence of unique expansions with respect to Kakeya sequences.

Bo Wang (Sun Yat-Sen University, Jiaying University)
2025-10-21 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/154/
Title: Ramírez’s Problems and Fibers on Well Approximable Set of Systems of Affine Forms (https://researchseminars.org/talk/OWNS/154/)
by Bo Wang (Sun Yat-Sen University, Jiaying University) as part of One World Numeration seminar

Abstract
In this talk, we show that badly approximable matrices are exactly those that, for every inhomogeneous parameter, cannot be inhomogeneous approximated at every monotone divergent rate, which generalizes Ramírez's result (2018). We also establish some metrical results of the fibers on well approximable set of systems of affine forms, which gives answers to three of Ramírez's problems (2018). Furthermore, we prove that badly approximable systems are exactly those that for each monotone convergent rate $\psi$ cannot be approximated at $\psi$. Moreover, we study the topological structure of the set of approximation functions. This is a joint work with Bing Li.

Colin Faverjon (CNRS, Université de Picardie Jules Verne)
2025-11-04 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/155/
Title: Writing numbers in multiple bases: the viewpoint of finite automata (https://researchseminars.org/talk/OWNS/155/)
by Colin Faverjon (CNRS, Université de Picardie Jules Verne) as part of One World Numeration seminar

Abstract
Although binary and decimal representations of numbers coexist seamlessly in our digital world, these changes of basis conceal profound mysteries. Consider, for example, the following statements, both currently out of reach:

- The date when Trump will leave the US presidency appears in the decimal expansion of every sufficiently large power of 2;

- The real number whose binary expansion is the characteristic sequence of powers of 3 contains the pattern 1312 infinitely often in its decimal expansion.

Both statements rely on the heuristic that expansions in multiplicatively independent bases (such as 2 and 10) should share no common structure. Furstenberg captured this heuristic through a series of results and conjectures concerning the joint behavior of the dynamical systems ×p and ×q on the torus.

In this talk, we approach this question from the perspective pioneered by Turing, Hartmanis, Stearns, and Cobham: that of computational complexity. Powers of 2 are particularly easy to recognize from their base-2 expansion—a task achievable by a finite automaton. Cobham's theorem then implies that no automaton can recognize them from their decimal expansion. Similarly, one can readily construct a finite automaton with output that produces the binary expansion of the real number introduced above. Whether there exists another automaton producing its decimal expansion remained open until recently.

In this talk, we present how this question has been solved using a transcendence method known as Mahler's method. While this approach yields a new proof and an algebraic generalization of Cobham's theorem, its main contribution is the following statement: no irrational real number has expansions in two multiplicatively independent bases that can both be produced by finite automata.

Paul Glendinning (University of Manchester)
2025-11-18 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/156/
Title: Positive Hausdorff dimension for the survivor set and transitions to chaos for piecewise smooth maps (https://researchseminars.org/talk/OWNS/156/)
by Paul Glendinning (University of Manchester) as part of One World Numeration seminar

Abstract
We consider two related problems. The transition to chaos in the sense of positive topological entropy for one-dimensional piecewise smooth maps, and the transition to positive Hausdorff dimension for the survivor set of associated open maps. We describe an iterative process that determines the boundaries of positive topological entropy (resp. positive Hausdorff dimension). The boundary can then be characterised via substitution sequences that generalise the Thue-Morse sequence for continuous maps of the interval. This work is joint with Clément Hege.

Michał Rams (IM PAN)
2025-12-16 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/157/
Title: The entropy of Lyapunov-optimizing measures for some matrix cocycles (https://researchseminars.org/talk/OWNS/157/)
by Michał Rams (IM PAN) as part of One World Numeration seminar

Abstract
Consider a simple (to formulate...) mathematical object: you are given a finite collection of matrices $A_i\in GL(2,\mathbb R); i=1,\ldots,k$ and you are allowed to multiply them, in any order. The notion you are interested in is the exponential rate of speed of growth of the norm: given $\omega\in \{1,\ldots,k\}^{\mathbb N}$, let
\[
\lambda(\omega) = \lim_{n\to\infty} \frac 1n \log ||A_{\omega_n} \cdot \ldots \cdot A_{\omega_1}||.
\]
This object has many names, in dynamical systems we call it the Lyapunov exponent.

We are in particular interested in the set of those $\omega$'s that give the extremal (maximal, minimal) value of the Lyapunov exponent. A long-standing conjecture states that for a generic matrix collection those sets ought to be {\it small}, in some sense. In the result I will present we (Jairo Bochi and me) are proving that for certain open set of collections of matrices those $\omega$'s that maximize/minimize Lyapunov exponent have zero topological entropy.

Shu-Qin Zhang (Zhengzhou University)
2025-12-02 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/158/
Title: When the conformal dimension of a self-affine sponge of Lalley type is zero (https://researchseminars.org/talk/OWNS/158/)
by Shu-Qin Zhang (Zhengzhou University) as part of One World Numeration seminar

Abstract
A compact metric space $X$ is uniformly disconnected if there exists $\delta_0$ such that there is no $\delta_0$-sequence, which is a sequence of points $(x_0,x_1,\dots,x_n)$ satisfying $\rho(x_{i-1},x_{i})\leq \delta_0 \rho(x_0,x_n)$ for all $1\leq i\leq n$. We present two main results. First, we give a necessary and sufficient condition for a diagonal self-affine sponge of Lalley-Gatzouras type to be uniformly disconnected. Second, we show that $K$ is uniformly disconnected if and only if the conformal dimension of $K$ is $0$.

Joaco Prandi (University of Waterloo)
2026-01-13 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/159/
Title: When the weak separation condition implies the generalize finite type in $\mathbb{R}^d$ (https://researchseminars.org/talk/OWNS/159/)
by Joaco Prandi (University of Waterloo) as part of One World Numeration seminar

Abstract
Let $S$ be an iterated function system with full support. Under some restrictions on the allowable rotations, we will show that $S$ satisfies the weak separation condition if and only if it satisfies the generalized finite-type condition. To do this, we will extend the notion of net intervals from $\mathbb{R}$ to $\mathbb{R}^d$. If time allows, we will also use net intervals to calculate the local dimension of a self-similar measure with the finite-type condition and full support. This talk is based on joint work with Kevin G. Hare.

Henri Cohen (Université de Bordeaux)
2026-01-27 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/160/
Title: Continued Fractions and Irrationality Measures for Chowla-Selberg Gamma Quotients (https://researchseminars.org/talk/OWNS/160/)
by Henri Cohen (Université de Bordeaux) as part of One World Numeration seminar

Abstract
We give 39 rapidly convergent continued fractions for Chowla--Selberg gamma quotients, and deduce good irrationality measures for 20 of them, including for $\operatorname{CS}(-3)=(\Gamma(1/3)/\Gamma(2/3))^3$, for $a^{1/4}\operatorname{CS}(-4)=a^{1/4}(\Gamma(1/4)/\Gamma(3/4))^2$ with $a=12$ and $a=1/5$, and for $\operatorname{CS}(-7)=\Gamma(1/7)\Gamma(2/7)\Gamma(4/7)/(\Gamma(3/7)\Gamma(5/7)\Gamma(6/7))$.
These appear to be the first proved and reasonable irrationality measures for
gamma quotients.

Junnosuke Koizumi (RIKEN iTHEMS)
2026-02-10 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/161/
Title: Irrationality Sequences (https://researchseminars.org/talk/OWNS/161/)
by Junnosuke Koizumi (RIKEN iTHEMS) as part of One World Numeration seminar

Abstract
Sometimes one can prove the irrationality of the sum of reciprocals of a sequence of positive integers using only information about the growth rate of the sequence. Erdős and Straus introduced the notion of an irrationality sequence in order to isolate nontrivial aspects of this relationship. Despite its elementary formulation, the theory of irrationality sequences still contains many open problems. For instance, the question of whether $2^{2^n}$ is a (type 2) irrationality sequence is a particularly interesting open problem. Recently, Kovač and Tao obtained several interesting results on the asymptotic behavior of irrationality sequences. We study sums of reciprocals of doubly exponential sequences and show, among other results, that there are at most countably many real numbers $a>1$ for which $a^{2^n}$ is a (type 2) irrationality sequence. We also explain how such questions are related to certain greedy Egyptian fraction expansions.

Pascal Jelinek (Montanuniversität Leoben)
2026-02-24 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/162/
Title: Ratio of sum of digits functions in two bases (https://researchseminars.org/talk/OWNS/162/)
by Pascal Jelinek (Montanuniversität Leoben) as part of One World Numeration seminar

Abstract
In 2019 La Bretèche, Stoll and Tenenbaum showed that the ratio of the sum of digits function $s_p(n)/s_q(n)$ of two multiplicatively independent bases $p$ and $q$ is dense in $\mathbb{Q}^+$. Spiegelhofer proved that when $p = 2$ and $q = 3$, the ratio 1 is attained infinitely many times, which he extended jointly with Drmota to arbitrary values in $\mathbb{Q}^+$. In this talk, I generalize this result further, showing that for two arbitrary multiplicatively independent bases, $s_p(n)/s_q(n)$ attains every value in $\mathbb{Q}^+$ infinitely many times.

Shintaro Suzuki (Tokyo Gakugei University)
2026-03-10 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/163/
Title: Hausdorff dimension of digit frequency sets for beta-expansions and a generalization of the Hata-Yamaguchi formula (https://researchseminars.org/talk/OWNS/163/)
by Shintaro Suzuki (Tokyo Gakugei University) as part of One World Numeration seminar

Abstract
We consider the digit frequency set of the digit 1 for beta-expansions in the case of $1<\beta\leq2$ and give an exact formula for its Hausdorff dimension via transfer operator method. As a related topic,
we introduce a generalization of the Lebesgue singular function and show that
it satisfies a version of the Hata-Yamaguchi formula, which yields a Takagi-like function for beta-expansions with the base $1<\beta<2$.

Yuru Zou (Shenzhen University)
2026-03-24 13:00 - 14:00 | https://researchseminars.org/talk/OWNS/164/
Title: Hausdorff dimension of double base expansions and binary shifts with a hole (https://researchseminars.org/talk/OWNS/164/)
by Yuru Zou (Shenzhen University) as part of One World Numeration seminar

Abstract
For two real bases $q_0, q_1 > 1$, a binary sequence $i_1 i_2 \cdots \in \{0,1\}^\infty$ is called the $(q_0,q_1)$-expansion of the number

\[
\pi_{q_0,q_1}(i_1 i_2 \cdots) = \sum_{k=1}^\infty \frac{i_k}{q_{i_1} \cdots q_{i_k}}.
\]
Let $\mathcal{U}_{q_0,q_1}$ denote the set of all real numbers having a unique $(q_0,q_1)$-expansion. When the two bases coincide, i.e., $q_0 = q_1 = q$, it was shown by Allaart and Kong (2019) that the Hausdorff dimension of the univoque set $\mathcal{U}_{q,q}$ varies continuously in $q$, building on earlier work of Komornik, Kong, and Li (2017). In this talk, we will derive explicit formulas for the Hausdorff dimension of $\mathcal{U}_{q_0,q_1}$ and for the topological entropy of the associated subshift for arbitrary $q_0, q_1 > 1$. We will also establish the continuity of these quantities as functions of the pair $(q_0,q_1)$.

Vítězslav Kala (Charles University)
2026-04-07 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/165/
Title: Using geometric continued fractions for universal quadratic forms (https://researchseminars.org/talk/OWNS/165/)
by Vítězslav Kala (Charles University) as part of One World Numeration seminar

Abstract
Already in the 18th century, Lagrange proved that every positive integer can be expressed as the sum of four squares of integers. Nowadays we say that the sum of four squares is a universal quadratic form. In the friendly and accessible talk, I’ll discuss some results on universal quadratic forms over Z and over totally real number fields. In particular, I'll focus on the role of continued fractions in many of the advances in the area over the last 10 years. First, I'll talk about the connection between classical continued fractions and quadratic forms over real quadratic fields. Then I'll turn to the more complicated situation of higher degree fields where we use geometric continued fractions as developed by Klein, Arnold, Karpenkov and many others. The talk is based on recent joint works with Valentin Blomer, Siu Hang Man, Magdalena Tinkova, Robin Visser, and Pavlo Yatsyna.

Jamie Walton (University of Nottingham)
2026-04-21 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/166/
Title: Geometric recognisability for FLC patterns (https://researchseminars.org/talk/OWNS/166/)
by Jamie Walton (University of Nottingham) as part of One World Numeration seminar

Abstract
Informally, recognisability of a substitution (or ‘inflate, subdivide’) rule regards its invertibility. Beyond the classical result of Mossé, highly general recognisability results have recently been established in the symbolic setting, even going beyond substitutions to S-adic sequences. In this talk, I will introduce a notion of a geometric tiling, point set or generalised ‘pattern’ of Euclidean space being substitutional with respect to an inflation map, in terms of two basic relations from Aperiodic Order: local derivability and local indistinguishability. These are analogues, from Symbolic Dynamics, of being related by sliding block codes and being in the same orbit closure, respectively. In the translational finite local complexity (FLC) case (e.g., tilings with finitely many tile types, meeting in finitely many ways up to translation), we give a formula for the number of pre-images of a pattern under substitution in terms of its group of translational periods. In particular, for a suitable power of substitution, the non-periodic tilings are precisely those with unique pre-images under substitution. Many techniques used are similar to those in Solomyak’s unique composition result, although we require no minimality (primitivity) assumption. If time permits, I will explain the motivation for this work, on the question of when a cut and project scheme produces substitutional patterns.

Gandhar Joshi (The Open University)
2026-05-05 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/167/
Title: Hiccup sequences, a Kimberling's conjecture, and Dumont-Thomas numeration systems (https://researchseminars.org/talk/OWNS/167/)
by Gandhar Joshi (The Open University) as part of One World Numeration seminar

Abstract
This is part of a joint work with Robbert Fokkink (TU Delft). We generalise the self-referential sequences introduced by Benoit Cloitre in 2003 on OEIS under the umbrella term ‘hiccup sequences’ with a direct skeletal influence from a ‘remarkable’ paper by Dekking, Bosma, and Steiner (2018) describing one such sequence in five very interesting ways. In this talk, we begin with one such way that uses morphisms. Using the Dumont-Thomas numeration system (DTNS) associated with the morphism, we prove a Kimberling’s conjecture on the OEIS about the bounds of a difference sequence related to one such sequence. There is also some use of Walnut software for Ollinger provides a tool that converts the DTNS into a set of Walnut-readable automata.

Bernat Espigulé (Universitat de Girona)
2026-06-16 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/168/
Title: Restricted polynomial roots, reciprocal power series, and finite capture for collinear affine IFS (https://researchseminars.org/talk/OWNS/168/)
by Bernat Espigulé (Universitat de Girona) as part of One World Numeration seminar

Abstract
I will discuss how roots of polynomials with coefficients in a finite symmetric integer digit set can be studied through a family of affine iterated function systems whose first-level pieces are centered on a line, evenly spaced, and symmetric with respect to the origin. After passing to reciprocal power series, the closure of the relevant root sets outside the unit disk can be interpreted as a connectedness locus: zeros of such power series describe overlaps in the attractor. This viewpoint is related to the theory of planar self-affine tiles with collinear digit sets, as studied by Akiyama, Loridant, Thuswaldner, and the broader self-affine-tile and number-system literature.

The main part of the talk will explain a finite-capture procedure for the non-real part of the locus, based on successive geometric enclosures. In a precise parameter range, this turns an infinite analytic problem into a finite geometric one. I will emphasize examples and pictures, with only minimal formulas.

This is based on joint work with David Juher, building on earlier joint work with David Juher and Joan Saldaña.

Toghrul Karimov (MPI-SWS, IRIF)
2026-06-30 12:00 - 13:00 | https://researchseminars.org/talk/OWNS/169/
Title: Automata on S-adic words (https://researchseminars.org/talk/OWNS/169/)
by Toghrul Karimov (MPI-SWS, IRIF) as part of One World Numeration seminar

Abstract
O. Carton and W. Thomas gave, in 2002, an algorithm for deciding whether a given automaton $A$ over infinite words accept a given morphic word $u$. Together with V. Berthé and M. Vahanwala, we study the same automaton acceptance problem in the more general setting of $S$-adic words. Among other results, we show how to compute, given a set $S$ of substitutions and an automaton $A$, an automaton $B$ that accepts a sequence $s$ over $S$ if and only if $s$ directs a word accepted by $A$. Thus we are able to completely answer questions of the form "Which Sturmian words $u$ are accepted by a given automaton $A$?" In particular, we show that whether $A$ accepts $u$ is completely determined by the first $N$ (that depends only on $A$) partial quotients of the slope of $u$. Our main tools are monoids and a new (?) structure theorem for $S$-adic expansions.
