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BEGIN:VEVENT
SUMMARY:Narad Rampersad (University of Winnipeg)
DTSTART;VALUE=DATE-TIME:20200505T123000Z
DTEND;VALUE=DATE-TIME:20200505T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/1
DESCRIPTION:Title: Ost
rowski numeration and repetitions in words\nby Narad Rampersad (Univer
sity of Winnipeg) as part of One World Numeration seminar\n\n\nAbstract\nO
ne 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 quan
tity (called the *repetition threshold*) for certain families of infinite
words. For example\, it is known that the repetition threshold for Sturmi
an words is 2+phi\, and this value is reached by the Fibonacci word. Rece
ntly\, this problem has been studied for *balanced words* (which generaliz
e Sturmian words) and *rich words*. The infinite words constructed to res
olve this problem can be defined in terms of the Ostrowski-numeration syst
em for certain continued-fraction expansions. They can be viewed as *Ostr
owski-automatic* sequences\, where we generalize the notion of *k-automati
c sequence* from the base-k numeration system to the Ostrowski numeration
system.\n
LOCATION:https://researchseminars.org/talk/OWNS/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Solomyak (University of Bar-Ilan)
DTSTART;VALUE=DATE-TIME:20200519T123000Z
DTEND;VALUE=DATE-TIME:20200519T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/2
DESCRIPTION:Title: On
singular substitution Z-actions\nby Boris Solomyak (University of Bar-
Ilan) as part of One World Numeration seminar\n\n\nAbstract\nWe consider p
rimitive aperiodic substitutions on $d$ letters and the spectral propertie
s of associated dynamical systems. In an earlier work we introduced a spec
tral cocycle\, related to a kind of matrix Riesz product\, which extends t
he (transpose) substitution matrix to the $d$-dimensional torus. The asymp
totic 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. \n\nThis is a joint work with A. Bufetov.\n
LOCATION:https://researchseminars.org/talk/OWNS/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Olivier Carton (Université de Paris)
DTSTART;VALUE=DATE-TIME:20200512T123000Z
DTEND;VALUE=DATE-TIME:20200512T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/3
DESCRIPTION:Title: Pre
servation of normality by selection\nby Olivier Carton (Université de
Paris) as part of One World Numeration seminar\n\n\nAbstract\nWe first re
call Agafonov's theorem which states that finite state selection preserves
normality. We also give two slight extensions of this result to non-obliv
ious selection and suffix selection. We also propose a similar statement i
n the more general setting of shifts of finite type by defining selections
which are compatible with the shift.\n
LOCATION:https://researchseminars.org/talk/OWNS/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Célia Cisternino (University of Liège)
DTSTART;VALUE=DATE-TIME:20200526T123000Z
DTEND;VALUE=DATE-TIME:20200526T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/4
DESCRIPTION:Title: Erg
odic behavior of transformations associated with alternate base expansions
\nby Célia Cisternino (University of Liège) as part of One World Num
eration seminar\n\n\nAbstract\nWe consider a p-tuple of real numbers great
er than 1\, $\\boldsymbol{\\beta} = (\\beta_1\,\\dots\,\\beta_p)$\, called
an alternate base\, to represent real numbers. Since these representation
s generalize the 𝛽-representation introduced by Rényi in 1958\, a lot
of questions arise. In this talk\, we will study the transformation genera
ting the alternate base expansions (greedy representations). First\, we wi
ll 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.
\n\nThis is a joint work with Émilie Charlier and Karma Dajani.\n
LOCATION:https://researchseminars.org/talk/OWNS/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simon Baker (University of Birmingham)
DTSTART;VALUE=DATE-TIME:20200609T123000Z
DTEND;VALUE=DATE-TIME:20200609T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/6
DESCRIPTION:Title: Equ
idistribution results for self-similar measures\nby Simon Baker (Unive
rsity of Birmingham) as part of One World Numeration seminar\n\n\nAbstract
\nA 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 t
alk I will discuss an analogue of this statement that holds for fractal me
asures. 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 i
s with respect to the natural measure on $C+t$.\n
LOCATION:https://researchseminars.org/talk/OWNS/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlos Matheus (Ecole Polytechnique)
DTSTART;VALUE=DATE-TIME:20200616T123000Z
DTEND;VALUE=DATE-TIME:20200616T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/7
DESCRIPTION:Title: App
roximations of the Lagrange and Markov spectra\nby Carlos Matheus (Eco
le Polytechnique) as part of One World Numeration seminar\n\n\nAbstract\nT
he Lagrange and Markov spectra are closed subsets of the positive real num
bers defined in terms of diophantine approximations. Their topological str
uctures are quite involved: they begin with an explicit discrete subset ac
cumulating 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 algori
thms to approximate (in Hausdorff topology) these spectra.\n
LOCATION:https://researchseminars.org/talk/OWNS/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niels Langeveld (Leiden University)
DTSTART;VALUE=DATE-TIME:20200630T123000Z
DTEND;VALUE=DATE-TIME:20200630T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/9
DESCRIPTION:Title: Con
tinued fractions with two non integer digits\nby Niels Langeveld (Leid
en University) as part of One World Numeration seminar\n\n\nAbstract\nIn t
his talk\, we will look at a family of continued fraction expansions for w
hich the digits in the expansions can attain two different (typically non-
integer) values\, named $\\alpha_1$ and $\\alpha_2$ with $\\alpha_1 \\alph
a_2 \\le 1/2$. If $\\alpha_1 \\alpha_2 < 1/2$ we can associate a dynamical
system to these expansions with a switch region and therefore with lazy a
nd greedy expansions. We will explore the parameter space and highlight ce
rtain values for which we can construct the natural extension (such as a f
amily for which the lowest digit cannot be followed by itself). We end the
talk with a list of open problems.\n
LOCATION:https://researchseminars.org/talk/OWNS/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hajime Kaneko (University of Tsukuba)
DTSTART;VALUE=DATE-TIME:20200707T123000Z
DTEND;VALUE=DATE-TIME:20200707T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/10
DESCRIPTION:Title: An
alogy of Lagrange spectrum related to geometric progressions\nby Hajim
e Kaneko (University of Tsukuba) as part of One World Numeration seminar\n
\n\nAbstract\nClassical Lagrange spectrum is defined by Diophantine approx
imation properties of arithmetic progressions. The theory of Lagrange spec
trum is related to number theory and symbolic dynamics. In our talk we int
roduce significantly analogous results of Lagrange spectrum in uniform dis
tribution theory of geometric progressions. In particular\, we discuss the
geometric sequences whose common ratios are Pisot numbers. For studying t
he fractional parts of geometric sequences\, we introduce certain numerati
on system. \n\nThis talk is based on a joint work with Shigeki Akiyama.\n
LOCATION:https://researchseminars.org/talk/OWNS/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Attila Pethő (University of Debrecen)
DTSTART;VALUE=DATE-TIME:20200714T123000Z
DTEND;VALUE=DATE-TIME:20200714T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/11
DESCRIPTION:Title: On
diophantine properties of generalized number systems - finite and periodi
c representations\nby Attila Pethő (University of Debrecen) as part o
f One World Numeration seminar\n\n\nAbstract\nIn 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 repre
sentation of rational integers. We prove under natural assumptions that th
ere 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 pol
ynomials at rational integers.\n
LOCATION:https://researchseminars.org/talk/OWNS/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Derong Kong (Chongqing University)
DTSTART;VALUE=DATE-TIME:20200623T123000Z
DTEND;VALUE=DATE-TIME:20200623T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/12
DESCRIPTION:Title: Un
ivoque bases of real numbers: local dimension\, Devil's staircase and isol
ated points\nby Derong Kong (Chongqing University) as part of One Worl
d Numeration seminar\n\n\nAbstract\nGiven a positive integer $M$ and a rea
l number $x$\, let $U(x)$ be the set of all bases $q \\in (1\,M+1]$ such t
hat $x$ 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 a
lso determine the critical values and the topological structure of $U(x)$.
\n
LOCATION:https://researchseminars.org/talk/OWNS/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bill Mance (Adam Mickiewicz University in Poznań)
DTSTART;VALUE=DATE-TIME:20200901T123000Z
DTEND;VALUE=DATE-TIME:20200901T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/13
DESCRIPTION:Title: Ho
tspot Lemmas for Noncompact Spaces\nby Bill Mance (Adam Mickiewicz Uni
versity in Poznań) as part of One World Numeration seminar\n\n\nAbstract\
nWe will explore a correction of several previously claimed generalization
s 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 m
ore work in recent years focused on studying various generalizations of th
e concept of a normal number to numeration systems with infinite digit set
s (for example\, various continued fraction expansions\, the Lüroth serie
s expansion and its generalizations\, and so on). Also\, highlighting this
(elementary) mistake may be helpful for those looking to study these nume
ration systems further and wishing to avoid some common pitfalls.\n
LOCATION:https://researchseminars.org/talk/OWNS/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bing Li (South China University of Technology)
DTSTART;VALUE=DATE-TIME:20200908T123000Z
DTEND;VALUE=DATE-TIME:20200908T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/14
DESCRIPTION:Title: So
me fractal problems in beta-expansions\nby Bing Li (South China Univer
sity of Technology) as part of One World Numeration seminar\n\n\nAbstract\
nFor greedy beta-expansions\, we study some fractal sets of real numbers w
hose orbits under beta-transformation share some common properties. For ex
ample\, the partial sum of the greedy beta-expansion converges with the sa
me order\, the orbit is not dense\, the orbit is always far from that of a
nother point etc. The usual tool is to approximate the beta-transformation
dynamical system by Markov subsystems. We also discuss the similar proble
ms for intermediate beta-expansions.\n
LOCATION:https://researchseminars.org/talk/OWNS/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jeffrey Shallit (University of Waterloo)
DTSTART;VALUE=DATE-TIME:20200915T123000Z
DTEND;VALUE=DATE-TIME:20200915T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/15
DESCRIPTION:Title: La
zy Ostrowski Numeration and Sturmian Words\nby Jeffrey Shallit (Univer
sity of Waterloo) as part of One World Numeration seminar\n\n\nAbstract\nI
n this talk I will discuss a new connection between the so-called "lazy Os
trowski" numeration system\, and periods of the prefixes of Sturmian chara
cteristic words. I will also give a relationship between periods and the s
o-called "initial critical exponent". This builds on work of Frid\, Berth
é-Holton-Zamboni\, Epifanio-Frougny-Gabriele-Mignosi\, and others\, and i
s joint work with Narad Rampersad and Daniel Gabric.\n
LOCATION:https://researchseminars.org/talk/OWNS/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yotam Smilansky (Rutgers University)
DTSTART;VALUE=DATE-TIME:20200922T123000Z
DTEND;VALUE=DATE-TIME:20200922T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/16
DESCRIPTION:Title: Mu
ltiscale Substitution Tilings\nby Yotam Smilansky (Rutgers University)
as part of One World Numeration seminar\n\n\nAbstract\nMultiscale substit
ution tilings are a new family of tilings of Euclidean space that are gene
rated by multiscale substitution rules. Unlike the standard setup of subst
itution tilings\, which is a basic object of study within the aperiodic or
der community and includes examples such as the Penrose and the pinwheel t
ilings\, multiple distinct scaling constants are allowed\, and the definin
g process of inflation and subdivision is a continuous one. Under a certai
n irrationality assumption on the scaling constants\, this construction gi
ves rise to a new class of tilings\, tiling spaces and tiling dynamical sy
stem\, which are intrinsically different from those that arise in the stan
dard setup. In the talk I will describe these new objects and discuss vari
ous structural\, geometrical\, statistical and dynamical results. Based on
joint work with Yaar Solomon.\n
LOCATION:https://researchseminars.org/talk/OWNS/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marta Maggioni (Leiden University)
DTSTART;VALUE=DATE-TIME:20200929T123000Z
DTEND;VALUE=DATE-TIME:20200929T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/17
DESCRIPTION:Title: Ra
ndom matching for random interval maps\nby Marta Maggioni (Leiden Univ
ersity) as part of One World Numeration seminar\n\n\nAbstract\nIn this tal
k we extend the notion of matching for deterministic transformations to ra
ndom matching for random interval maps. For a large class of piecewise aff
ine random systems of the interval\, we prove that this property of random
matching implies that any invariant density of a stationary measure is pi
ecewise constant. We provide examples of random matching for a variety of
families of random dynamical systems\, that includes generalised beta-tran
sformations\, continued fraction maps and a family of random maps producin
g signed binary expansions. We finally apply the property of random matchi
ng and its consequences to this family to study minimal weight expansions.
\nBased on a joint work with Karma Dajani and Charlene Kalle.\n
LOCATION:https://researchseminars.org/talk/OWNS/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Francesco Veneziano (University of Genova)
DTSTART;VALUE=DATE-TIME:20201006T123000Z
DTEND;VALUE=DATE-TIME:20201006T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/18
DESCRIPTION:Title: Fi
niteness and periodicity of continued fractions over quadratic number fiel
ds\nby Francesco Veneziano (University of Genova) as part of One World
Numeration seminar\n\n\nAbstract\nWe consider continued fractions with pa
rtial quotients in the ring of integers of a quadratic number field $K$\;
a particular example of these continued fractions is the $\\beta$-continue
d fraction introduced by Bernat. We show that for any quadratic Perron num
ber $\\beta$\, 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 $\\beta$-con
tinued fraction represents finitely all elements of the quadratic field $\
\mathbb{Q}(\\beta)$\, thus answering questions of Rosen and Bernat. \nBase
d on a joint work with Zuzana Masáková and Tomáš Vávra.\n
LOCATION:https://researchseminars.org/talk/OWNS/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kan Jiang (Ningbo University)
DTSTART;VALUE=DATE-TIME:20201013T123000Z
DTEND;VALUE=DATE-TIME:20201013T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/19
DESCRIPTION:Title: Re
presentations of real numbers on fractal sets\nby Kan Jiang (Ningbo Un
iversity) as part of One World Numeration seminar\n\n\nAbstract\nThere are
many approaches which can represent real numbers. For instance\, the $\\b
eta$-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-thi
rd 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 re
sults in terms of some fractal sets.\n
LOCATION:https://researchseminars.org/talk/OWNS/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paul Surer (University of Natural Resources and Life Sciences\, Vi
enna)
DTSTART;VALUE=DATE-TIME:20201020T123000Z
DTEND;VALUE=DATE-TIME:20201020T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/20
DESCRIPTION:Title: Re
presentations for complex numbers with integer digits\nby Paul Surer (
University of Natural Resources and Life Sciences\, Vienna) as part of One
World Numeration seminar\n\n\nAbstract\nIn 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 inte
ger digits. Our concepts fits into the framework of the recently published
rotational beta-expansions. But we also establish relations with piecewis
e affine maps of the torus and with shift radix systems.\n
LOCATION:https://researchseminars.org/talk/OWNS/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mélodie Andrieu (Aix-Marseille University)
DTSTART;VALUE=DATE-TIME:20201027T133000Z
DTEND;VALUE=DATE-TIME:20201027T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/21
DESCRIPTION:Title: A
Rauzy fractal unbounded in all directions of the plane\nby Mélodie An
drieu (Aix-Marseille University) as part of One World Numeration seminar\n
\n\nAbstract\nUntil 2001 it was believed that\, as for Sturmian words\, th
e imbalance of Arnoux-Rauzy words was bounded - or at least finite. Cassai
gne\, Ferenczi and Zamboni disproved this conjecture by constructing an Ar
noux-Rauzy word with infinite imbalance\, i.e. a word whose broken line de
viates regularly and further and further from its average direction. Today
\, we hardly know anything about the geometrical and topological propertie
s of these unbalanced Rauzy fractals. The Oseledets theorem suggests that
these fractals are contained in a strip of the plane: indeed\, if the Lyap
unov exponents of the matricial product associated with the word exist\, o
ne of these exponents at least is nonpositive since their sum equals zero.
This talk aims at disproving this belief.\n
LOCATION:https://researchseminars.org/talk/OWNS/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tomáš Vávra (University of Waterloo)
DTSTART;VALUE=DATE-TIME:20201103T133000Z
DTEND;VALUE=DATE-TIME:20201103T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/22
DESCRIPTION:Title: Di
stinct unit generated number fields and finiteness in number systems\n
by Tomáš Vávra (University of Waterloo) as part of One World Numeration
seminar\n\n\nAbstract\nA distinct unit generated field is a number field
K such that every algebraic integer of the field is a sum of distinct unit
s. In 2015\, Dombek\, Masáková\, and Ziegler studied totally complex qua
rtic fields\, leaving 8 cases unresolved. Because in this case there is on
ly one fundamental unit $u$\, their method involved the study of finitenes
s in positional number systems with base u and digits arising from the roo
ts of unity in $K$.\n \nFirst\, we consider a more general problem of posi
tional representations with base beta with an arbitrary digit alphabet $D$
. We will show that it is decidable whether a given pair $(\\beta\, D)$ al
lows eventually periodic or finite representations of elements of $O_K$.\n
\nWe are then able to prove the conjecture that the 8 remaining cases ind
eed are distinct unit generated.\n
LOCATION:https://researchseminars.org/talk/OWNS/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pieter Allaart (University of North Texas)
DTSTART;VALUE=DATE-TIME:20201110T133000Z
DTEND;VALUE=DATE-TIME:20201110T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/23
DESCRIPTION:Title: On
the smallest base in which a number has a unique expansion\nby Pieter
Allaart (University of North Texas) as part of One World Numeration semin
ar\n\n\nAbstract\nFor $x>0$\, 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 h
as infinitely many infinite level sets. I will describe an algorithm for n
umerically computing $f(x)$ that often gives the exact value in just a sma
ll 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 Ko
ng\, and builds on previous work by Kong (Acta Math. Hungar. 150(1):194--2
08\, 2016).\n
LOCATION:https://researchseminars.org/talk/OWNS/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jacques Sakarovitch (Irif\, CNRS\, and Télécom Paris)
DTSTART;VALUE=DATE-TIME:20201117T133000Z
DTEND;VALUE=DATE-TIME:20201117T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/24
DESCRIPTION:Title: Th
e carry propagation of the successor function\nby Jacques Sakarovitch
(Irif\, CNRS\, and Télécom Paris) as part of One World Numeration semina
r\n\n\nAbstract\nGiven any numeration system\, the carry propagation at an
integer $N$ is the number of digits that change between the representatio
n of $N$ and $N+1$. The carry propagation of the numeration system as a wh
ole is the average carry propagations at the first $N$ integers\, as $N$ t
ends to infinity\, if this limit exists. \n\nIn 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 sys
tems we consider and for which the limit exists. \n\nThe problem is less t
he computation of the carry propagation than the proof of its existence. W
e 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 te
chniques: combinatorial\, algebraic\, and ergodic\, each of them being rel
evant for different kinds of numeration systems. \n\nThis work has been pu
blished in Advances in Applied Mathematics 120 (2020). In this talk\, we s
hall focus on the algebraic and ergodic methods. \n\nJoint work with V. Be
rthé (Irif)\, Ch. Frougny (Irif)\, and M. Rigo (Univ. Liège).\n
LOCATION:https://researchseminars.org/talk/OWNS/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Barnsley (Australian National University)
DTSTART;VALUE=DATE-TIME:20201201T133000Z
DTEND;VALUE=DATE-TIME:20201201T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/25
DESCRIPTION:Title: Ri
gid fractal tilings\nby Michael Barnsley (Australian National Universi
ty) as part of One World Numeration seminar\n\n\nAbstract\nI will describe
recent work\, joint with Louisa Barnsley and Andrew Vince\, concerning a
symbolic approach to self-similar tilings. This approach uses graph-direct
ed iterated function systems to analyze both classical tilings and also ge
neralized tilings of what may be unbounded fractal subsets of $\\mathbb{R}
^n$. A notion of rigid tiling systems is defined. Our key theorem states t
hat when the system is rigid\, all the conjugacies of the tilings can be d
escribed explicitly. In the seminar I hope to prove this for the case of s
tandard IFSs.\n
LOCATION:https://researchseminars.org/talk/OWNS/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tanja Isabelle Schindler (Scuola Normale Superiore di Pisa)
DTSTART;VALUE=DATE-TIME:20201208T133000Z
DTEND;VALUE=DATE-TIME:20201208T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/26
DESCRIPTION:Title: Li
mit theorems on counting large continued fraction digits\nby Tanja Isa
belle Schindler (Scuola Normale Superiore di Pisa) as part of One World Nu
meration seminar\n\n\nAbstract\nWe establish a central limit theorem for c
ounting large continued fraction digits $(a_n)$\, that is\, we count occur
rences $\\{a_n>b_n\\}$\, where $(b_n)$ is a sequence of positive integers.
Our result improves a similar result by Philipp\, which additionally assu
mes that bn tends to infinity. Moreover\, we also show this kind of centra
l limit theorem for counting the number of occurrences entries such that t
he continued fraction entry lies between $d_n$ and $d_n(1+1/c_n)$ for give
n sequences $(c_n)$ and $(d_n)$. For such intervals we also give a refinem
ent of the famous Borel–Bernstein theorem regarding the event that the n
th 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 the
orem. This is joint work with Marc Kesseböhmer.\n
LOCATION:https://researchseminars.org/talk/OWNS/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lukas Spiegelhofer (Montanuniversität Leoben)
DTSTART;VALUE=DATE-TIME:20201215T133000Z
DTEND;VALUE=DATE-TIME:20201215T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/27
DESCRIPTION:Title: Th
e digits of $n+t$\nby Lukas Spiegelhofer (Montanuniversität Leoben) a
s part of One World Numeration seminar\n\n\nAbstract\nWe study the binary
sum-of-digits function $s_2$ under addition of a constant $t$.\nFor each i
nteger $k$\, we are interested in the asymptotic density $\\delta(k\,t)$ o
f integers $t$ such that $s_2(n+t)-s_2(n)=k$.\nIn this talk\, we consider
the following two questions. \n\n(1) Do we have \\[ c_t=\\delta(0\,t)+\\
delta(1\,t)+\\cdots>1/2? \\]\nThis is a conjecture due to T. W. Cusick (2
011). \n\n(2) What does the probability distribution defined by $k\\mapsto
\\delta(k\,t)$ look like?\n\nWe 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.\nOur second theorem states that $\\delta(j\,t)$ usually
behaves like a normal distribution\, which extends a result by Emme and Hu
bert (2018).\n\nThis is joint work with Michael Wallner (TU Wien).\n
LOCATION:https://researchseminars.org/talk/OWNS/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claire Merriman (Ohio State University)
DTSTART;VALUE=DATE-TIME:20210105T133000Z
DTEND;VALUE=DATE-TIME:20210105T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/28
DESCRIPTION:Title: $\
\alpha$-odd continued fractions\nby Claire Merriman (Ohio State Univer
sity) as part of One World Numeration seminar\n\n\nAbstract\nThe standard
continued fraction algorithm come from the Euclidean algorithm. We can als
o 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 dynamica
l system when we impose restrictions on the continued fraction expansion\,
such as finding the nearest odd integer instead of the floor? This talk w
ill focus on the case where we first restrict to odd integers\, then start
shifting the domain $[\\alpha-2\, \\alpha)$.\n \nThis talk is based on jo
int work with Florin Boca and animations done by Xavier Ding\, Gustav Jenn
etten\, and Joel Rozhon as part of an Illinois Geometry Lab project.\n
LOCATION:https://researchseminars.org/talk/OWNS/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tom Kempton (University of Manchester)
DTSTART;VALUE=DATE-TIME:20210119T133000Z
DTEND;VALUE=DATE-TIME:20210119T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/29
DESCRIPTION:Title: Be
rnoulli Convolutions and Measures on the Spectra of Algebraic Integers
\nby Tom Kempton (University of Manchester) as part of One World Numeratio
n seminar\n\n\nAbstract\nGiven an algebraic integer $\\beta$ and alphabet
$A=\\{-1\,0\,1\\}$\, the spectrum of $\\beta$ is the set \n$$\\Sigma(\\bet
a) :=\\bigg\\{\\sum_{i=1}^n a_i\\beta^i : n\\in\\mathbb N\, a_i\\in A\\big
g\\}.$$\nIn 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.\n\nIn this talk
we will define a random walk on the spectrum of $\\beta$ and show how\, w
ith appropriate renormalisation\, this leads to an infinite stationary mea
sure on the spectrum. This measure has local structure analagous to that o
f the spectrum itself. Furthermore\, this measure has deep links with the
Bernoulli convolution\, and in particular new criteria for the absolute co
ntinuity of Bernoulli convolutions can be stated in terms of the ergodic p
roperties of these measures.\n
LOCATION:https://researchseminars.org/talk/OWNS/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Carlo Carminati (Università di Pisa)
DTSTART;VALUE=DATE-TIME:20210126T133000Z
DTEND;VALUE=DATE-TIME:20210126T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/30
DESCRIPTION:Title: Pr
evalence of matching for families of continued fraction algorithms: old an
d new results\nby Carlo Carminati (Università di Pisa) as part of One
World Numeration seminar\n\n\nAbstract\nWe will give an overview of the p
henomenon 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.\n\nOur main focus will be the matching
property for the family of Ito-Tanaka continued fractions: we will discuss
the analogies with Nakada's case\n(such as prevalence of matching)\, but
also some unexpected features which are peculiar of this case.\n\nThe core
of the talk is about some recent results obtained in collaboration with N
iels Langeveld and Wolfgang Steiner.\n
LOCATION:https://researchseminars.org/talk/OWNS/30/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samuel Petite (Université de Picardie Jules Verne)
DTSTART;VALUE=DATE-TIME:20210202T133000Z
DTEND;VALUE=DATE-TIME:20210202T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/31
DESCRIPTION:Title: In
terplay between finite topological rank minimal Cantor systems\, $S$-adic
subshifts and their complexity\nby Samuel Petite (Université de Picar
die Jules Verne) as part of One World Numeration seminar\n\n\nAbstract\nTh
e family of minimal Cantor systems of finite topological rank includes Stu
rmian subshifts\, coding of interval exchange transformations\, odometers
and substitutive subshifts. They are known to have dynamical rigidity prop
erties. In a joint work with F. Durand\, S. Donoso and A. Maass\, we provi
de a combinatorial characterization of such subshifts in terms of S-adic s
ystems. This enables to obtain some links with the factor complexity funct
ion and some new rigidity properties depending on the rank of the system.\
n
LOCATION:https://researchseminars.org/talk/OWNS/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Clemens Müllner (TU Wien)
DTSTART;VALUE=DATE-TIME:20210209T133000Z
DTEND;VALUE=DATE-TIME:20210209T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/32
DESCRIPTION:Title: Mu
ltiplicative automatic sequences\nby Clemens Müllner (TU Wien) as par
t of One World Numeration seminar\n\n\nAbstract\nIt was shown by Mariusz L
emańczyk and the author that automatic sequences are orthogonal to bounde
d and aperiodic multiplicative functions. This is a manifestation of the d
isjointedness of additive and multiplicative structures. We continue this
path by presenting in this talk a complete classification of complex-value
d sequences which are both multiplicative and automatic. This shows that t
he intersection of these two worlds has a very special (and simple) form.
This is joint work with Mariusz Lemańczyk and Jakub Konieczny.\n
LOCATION:https://researchseminars.org/talk/OWNS/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gerardo González Robert (Universidad Nacional Autónoma de Méxic
o)
DTSTART;VALUE=DATE-TIME:20210216T133000Z
DTEND;VALUE=DATE-TIME:20210216T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/33
DESCRIPTION:Title: Go
od's Theorem for Hurwitz Continued Fractions\nby Gerardo González Rob
ert (Universidad Nacional Autónoma de México) as part of One World Numer
ation seminar\n\n\nAbstract\nIn 1887\, Adolf Hurwitz introduced a simple p
rocedure 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 com
plex 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 comp
lex version of I. J. Good's theorem on the Hausdorff dimension of the set
of real numbers whose regular continued fraction tends to infinity. Finall
y\, we will discuss some open problems.\n
LOCATION:https://researchseminars.org/talk/OWNS/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Seulbee Lee (Scuola Normale Superiore di Pisa)
DTSTART;VALUE=DATE-TIME:20210223T133000Z
DTEND;VALUE=DATE-TIME:20210223T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/34
DESCRIPTION:Title: Od
d-odd continued fraction algorithm\nby Seulbee Lee (Scuola Normale Sup
eriore di Pisa) as part of One World Numeration seminar\n\n\nAbstract\nThe
classical continued fraction gives the best approximating rational number
s of an irrational number. We define a new continued fraction\, say odd-od
d continued fraction\, which gives the best approximating rational numbers
whose numerators and denominators are odd. We see that a jump transformat
ion 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.\n
LOCATION:https://researchseminars.org/talk/OWNS/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vitaly Bergelson (Ohio State University)
DTSTART;VALUE=DATE-TIME:20210302T150000Z
DTEND;VALUE=DATE-TIME:20210302T160000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/35
DESCRIPTION:Title: No
rmal sets in $(\\mathbb{ℕ}\,+)$ and $(\\mathbb{N}\,\\times)$\nby Vit
aly Bergelson (Ohio State University) as part of One World Numeration semi
nar\n\n\nAbstract\nWe will start with discussing the general idea of a nor
mal set in a countable cancellative amenable semigroup\, which was introdu
ced and developed in the recent paper "A fresh look at the notion of norma
lity" (joint work with Tomas Downarowicz and Michał Misiurewicz). We will
move then to discussing and juxtaposing combinatorial and Diophantine pro
perties of normal sets in semigroups $(\\mathbb{ℕ}\,+)$ and $(\\mathbb{N
}\,\\times)$. We will conclude the lecture with a brief review of some int
eresting open problems.\n
LOCATION:https://researchseminars.org/talk/OWNS/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Natalie Priebe Frank (Vassar College)
DTSTART;VALUE=DATE-TIME:20210309T133000Z
DTEND;VALUE=DATE-TIME:20210309T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/36
DESCRIPTION:Title: Th
e flow view and infinite interval exchange transformation of a recognizabl
e substitution\nby Natalie Priebe Frank (Vassar College) as part of On
e World Numeration seminar\n\n\nAbstract\nA flow view is the graph of a me
asurable conjugacy between a substitution or S-adic subshift or tiling spa
ce and an exchange of infinitely many intervals in [0\,1]. The natural ref
ining 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 vie
w\, sequences are shown horizontally at a height given by their image unde
r conjugacy.\n\nIn this talk I'll explain how it all works and state some
results and questions. There will be pictures.\n
LOCATION:https://researchseminars.org/talk/OWNS/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandra Skripchenko (Higher School of Economics)
DTSTART;VALUE=DATE-TIME:20210316T133000Z
DTEND;VALUE=DATE-TIME:20210316T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/37
DESCRIPTION:Title: Do
uble rotations and their ergodic properties\nby Alexandra Skripchenko
(Higher School of Economics) as part of One World Numeration seminar\n\n\n
Abstract\nDouble rotations are the simplest subclass of interval translati
on mappings. A double rotation is of finite type if its attractor is an in
terval 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 si
mply a rotation. It is known due to Suzuki - Ito - Aihara and Bruin - Clar
k that double rotations of infinite type are defined by a subset of zero m
easure in the parameter set. We introduce a new renormalization procedure
on double rotations\, which is reminiscent of the classical Rauzy inductio
n. Using this renormalization we prove that the set of parameters which in
duce infinite type double rotations has Hausdorff dimension strictly small
er than 3. Moreover\, we construct a natural invariant measure supported o
n these parameters and show that\, with respect to this measure\, almost a
ll double rotations are uniquely ergodic. In my talk I plan to outline thi
s proof that is based on the recent result by Ch. Fougeron for simplicial
systems. I also hope to discuss briefly some challenging open questions an
d further research plans related to double rotations. \n\nThe talk is base
d on a joint work with Mauro Artigiani\, Charles Fougeron and Pascal Huber
t.\n
LOCATION:https://researchseminars.org/talk/OWNS/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Godofredo Iommi (Pontificia Universidad Católica de Chile)
DTSTART;VALUE=DATE-TIME:20210323T133000Z
DTEND;VALUE=DATE-TIME:20210323T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/38
DESCRIPTION:Title: Ar
ithmetic averages and normality in continued fractions\nby Godofredo I
ommi (Pontificia Universidad Católica de Chile) as part of One World Nume
ration seminar\n\n\nAbstract\nEvery real number can be written as a contin
ued 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 Hausdo
rff dimension of two types of sets: one of them defined in terms of arithm
etic averages of the digits in the expansion and the other related to (con
tinued fraction) normal numbers. In both cases\, the non compactness that
steams from the fact that we use countable many partial quotients in the c
ontinued fraction plays a fundamental role. Some of the results are joint
work with Thomas Jordan and others together with Aníbal Velozo.\n
LOCATION:https://researchseminars.org/talk/OWNS/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Drmota (TU Wien)
DTSTART;VALUE=DATE-TIME:20210330T123000Z
DTEND;VALUE=DATE-TIME:20210330T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/39
DESCRIPTION:Title: (L
ogarithmic) Densities for Automatic Sequences along Primes and Squares
\nby Michael Drmota (TU Wien) as part of One World Numeration seminar\n\n\
nAbstract\nIt is well known that the every letter $\\alpha$ of an automati
c sequence $a(n)$ has\na logarithmic density -- and it can be decided when
this logarithmic density is actually a density.\nFor example\, the letter
s $0$ and $1$ of the Thue-Morse sequences $t(n)$ have both frequences $1/2
$.\n[The Thue-Morse sequence is the binary sum-of-digits functions modulo
2.]\n\nThe purpose of this talk is to present a corresponding result for s
ubsequences of general\nautomatic sequences along primes and squares. This
is a far reaching generalization of two breakthrough\nresults of Mauduit
and Rivat from 2009 and 2010\, where they solved two conjectures by Gelfon
d\non the densities of $0$ and $1$ of $t(p_n)$ and $t(n^2)$ (where $p_n$ d
enotes the sequence of primes).\n\nMore technically\, one has to develop a
method to transfer density results for primitive automatic\nsequences to
logarithmic-density results for general automatic sequences. Then as an ap
plication\none can deduce that the logarithmic densities of any automatic
sequence along squares\n$(n^2)_{n\\geq 0}$ and primes $(p_n)_{n\\geq 1}$ e
xist and are computable.\nFurthermore\, if densities exist then they are (
usually) rational.\n
LOCATION:https://researchseminars.org/talk/OWNS/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Mitchell (University of Birmingham)
DTSTART;VALUE=DATE-TIME:20210413T123000Z
DTEND;VALUE=DATE-TIME:20210413T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/40
DESCRIPTION:Title: Me
asure theoretic entropy of random substitutions\nby Andrew Mitchell (U
niversity of Birmingham) as part of One World Numeration seminar\n\n\nAbst
ract\nRandom 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 t
heoretic 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. \n\nAs an application of our results\, we obtain closed form f
ormulas for the entropy of frequency measures for a wide range of random s
ubstitution 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 ent
ropy.\n\nThis talk is based on joint work with P. Gohlke\, D. Rust\, and T
. Samuel.\n
LOCATION:https://researchseminars.org/talk/OWNS/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ayreena Bakhtawar (La Trobe University)
DTSTART;VALUE=DATE-TIME:20210420T123000Z
DTEND;VALUE=DATE-TIME:20210420T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/41
DESCRIPTION:Title: Me
trical theory for the set of points associated with the generalized Jarnik
-Besicovitch set\nby Ayreena Bakhtawar (La Trobe University) as part o
f One World Numeration seminar\n\n\nAbstract\nFrom Lagrange's (1770) and L
egendre's (1808) results we conclude that to find good rational approximat
ions 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 f
raction 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 quoti
ent of $x$ and $T$ is the Gauss map. In this talk\, I will focus on determ
ining the Hausdorff dimension of the set of real numbers $x \\in [0\,1)$ s
uch that for any $m \\in \\mathbb{N}$ the following holds for infinitely m
any $n \\in \\mathbb{N}$: $a_{n+1}(x) a_{n+2}(x) \\cdots a_{n+m}(x) \\ge e
^{τ(x)\\\, (f(x)+⋯+f(T^{n-1}x))}$\, 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 incl
uding the famous Jarnik-Besicovitch theorem.\n
LOCATION:https://researchseminars.org/talk/OWNS/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Boris Adamczewski (CNRS\, Université Claude Bernard Lyon 1)
DTSTART;VALUE=DATE-TIME:20210427T123000Z
DTEND;VALUE=DATE-TIME:20210427T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/42
DESCRIPTION:Title: Ex
pansions of numbers in multiplicatively independent bases: Furstenberg's c
onjecture and finite automata\nby Boris Adamczewski (CNRS\, Universit
é Claude Bernard Lyon 1) as part of One World Numeration seminar\n\n\nAbs
tract\nIt is commonly expected that expansions of numbers in multiplicativ
ely independent bases\, such as 2 and 10\, should have no common structure
. However\, it seems extraordinarily difficult to confirm this naive heuri
stic principle in some way or another. In the late 1960s\, Furstenberg sug
gested a series of conjectures\, which became famous and aim to capture th
is 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 th
e language of automata theory to formulate some related problems that form
alize and express in a different way the same general heuristic. I will ex
plain how the latter can be solved thanks to some recent advances in Mahle
r’s method\; a method in transcendental number theory initiated by Mahle
r at the end of the 1920s. This a joint work with Colin Faverjon.\n
LOCATION:https://researchseminars.org/talk/OWNS/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tushar Das (University of Wisconsin - La Crosse)
DTSTART;VALUE=DATE-TIME:20210504T140000Z
DTEND;VALUE=DATE-TIME:20210504T150000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/43
DESCRIPTION:Title: Ha
usdorff Hensley Good & Gauss\nby Tushar Das (University of Wisconsin -
La Crosse) as part of One World Numeration seminar\n\n\nAbstract\nSeveral
participants of the One World Numeration Seminar (OWNS) will know Hensley
's haunting bounds (c. 1990) for the dimension of irrationals whose regula
r continued fraction expansion partial quotients are all at most N\; while
some might remember Good's great bounds (c. 1940) for the dimension of ir
rationals whose partial quotients are all at least N. We will report on re
latively 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\, dynami
cs\, or geometry. The talk will be accessible to students and beyond\, and
I hope to present a sampling of open questions and research directions th
at await exploration.\n
LOCATION:https://researchseminars.org/talk/OWNS/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Giulio Tiozzo (University of Toronto)
DTSTART;VALUE=DATE-TIME:20210511T123000Z
DTEND;VALUE=DATE-TIME:20210511T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/44
DESCRIPTION:Title: Th
e bifurcation locus for numbers of bounded type\nby Giulio Tiozzo (Uni
versity of Toronto) as part of One World Numeration seminar\n\n\nAbstract\
nWe define a family $B(t)$ of compact subsets of the unit interval which p
rovides a filtration of the set of numbers whose continued fraction expans
ion has bounded digits. This generalizes to a continuous family the well-k
nown sets of numbers whose continued fraction expansion is bounded above b
y a fixed integer. \n\nWe study how the set $B(t)$ changes as the paramete
r $t$ ranges in $[0\,1]$\, and describe precisely the bifurcations that oc
cur as the parameters change. Further\, we discuss continuity properties o
f the Hausdorff dimension of $B(t)$ and its regularity. \n\nFinally\, we e
stablish a precise correspondence between these bifurcations and \nthe bif
urcations for the classical family of real quadratic polynomials. \n\nJoin
t with C. Carminati.\n
LOCATION:https://researchseminars.org/talk/OWNS/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Joseph Vandehey (University of Texas at Tyler)
DTSTART;VALUE=DATE-TIME:20210518T123000Z
DTEND;VALUE=DATE-TIME:20210518T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/45
DESCRIPTION:Title: So
lved and unsolved problems in normal numbers\nby Joseph Vandehey (Univ
ersity of Texas at Tyler) as part of One World Numeration seminar\n\n\nAbs
tract\nWe 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 numb
ers\, normality in two different systems simultaneously\, normality seen t
hrough the lens of informational or logical complexity\, and more.\n
LOCATION:https://researchseminars.org/talk/OWNS/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Charles Fougeron (Université de Paris)
DTSTART;VALUE=DATE-TIME:20210525T123000Z
DTEND;VALUE=DATE-TIME:20210525T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/46
DESCRIPTION:Title: Dy
namics of simplicial systems and multidimensional continued fraction algor
ithms\nby Charles Fougeron (Université de Paris) as part of One World
Numeration seminar\n\n\nAbstract\nMotivated 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 generalisat
ions to study Diophantine approximations of vectors in higher dimensions.
Examples include Poincaré's algorithm introduced at the end of the 19th c
entury or those of Brun and Selmer in the middle of the 20th century. Sinc
e 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 converg
e to the right vector with an extra ergodic property. On the other hand\,
Nogueira demonstrated that the algorithm proposed by Poincaré almost neve
r converges. \n\nStarting from the classical case of Farey's algorithm\, w
hich is an "additive" version of Gauss's algorithm\, I will present a comb
inatorial point of view on these algorithms which allows to us to use a ra
ndom walk approach. In this model\, taking a random vector for the Lebesgu
e measure will correspond to following a random walk with memory in a labe
lled graph called symplicial system. The laws of probability for this rand
om 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 f
raction algorithm.\n
LOCATION:https://researchseminars.org/talk/OWNS/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bastián Espinoza (Université de Picardie Jules Verne and Univers
idad de Chile)
DTSTART;VALUE=DATE-TIME:20210601T123000Z
DTEND;VALUE=DATE-TIME:20210601T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/47
DESCRIPTION:Title: Au
tomorphisms and factors of finite topological rank systems\nby Bastiá
n Espinoza (Université de Picardie Jules Verne and Universidad de Chile)
as part of One World Numeration seminar\n\n\nAbstract\nFinite topological
rank systems are a type of minimal S-adic subshift that includes many of t
he classical minimal systems of zero entropy (e.g. linearly recurrent subs
hifts\, interval exchanges and some Toeplitz sequences). In this talk I am
going to present results concerning the number of automorphisms and facto
rs of systems of finite topological rank\, as well as closure properties o
f this class with respect to factors and related combinatorial operations.
\n
LOCATION:https://researchseminars.org/talk/OWNS/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shigeki Akiyama (University of Tsukuba)
DTSTART;VALUE=DATE-TIME:20210608T123000Z
DTEND;VALUE=DATE-TIME:20210608T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/48
DESCRIPTION:Title: Co
unting balanced words and related problems\nby Shigeki Akiyama (Univer
sity of Tsukuba) as part of One World Numeration seminar\n\n\nAbstract\nBa
lanced words and Sturmian words are ubiquitous and appear in the intersect
ion 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 interce
pt. Applying this idea\, we can obtain precise asymptotic formula for bala
nced words. The result is connected to some classical topics in number the
ory\, such as Farey fraction\, Riemann Hypothesis and Large sieve inequali
ty.\n
LOCATION:https://researchseminars.org/talk/OWNS/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sam Chow (University of Warwick)
DTSTART;VALUE=DATE-TIME:20210615T123000Z
DTEND;VALUE=DATE-TIME:20210615T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/49
DESCRIPTION:Title: Dy
adic approximation in the Cantor set\nby Sam Chow (University of Warwi
ck) as part of One World Numeration seminar\n\n\nAbstract\nWe investigate
the approximation rate of a typical element of the Cantor set by dyadic ra
tionals. This is a manifestation of the times-two-times-three phenomenon\,
and is joint work with Demi Allen and Han Yu.\n
LOCATION:https://researchseminars.org/talk/OWNS/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lingmin Liao (Université Paris-Est Créteil Val de Marne)
DTSTART;VALUE=DATE-TIME:20210622T123000Z
DTEND;VALUE=DATE-TIME:20210622T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/50
DESCRIPTION:Title: Si
multaneous Diophantine approximation of the orbits of the dynamical system
s x2 and x3\nby Lingmin Liao (Université Paris-Est Créteil Val de Ma
rne) as part of One World Numeration seminar\n\n\nAbstract\nWe study the s
ets of points whose orbits of the dynamical systems x2 and x3 simultaneous
ly 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 th
em is established under the abc conjecture. At the same time\, we also stu
dy the Diophantine approximation of the orbits of a diagonal matrix transf
ormation of a torus\, for which the properties of the (negative) beta tran
sformations are involved. This is a joint work with Bing Li\, Sanju Velani
and Evgeniy Zorin.\n
LOCATION:https://researchseminars.org/talk/OWNS/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Polina Vytnova (University of Warwick)
DTSTART;VALUE=DATE-TIME:20210629T123000Z
DTEND;VALUE=DATE-TIME:20210629T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/51
DESCRIPTION:Title: Ha
usdorff dimension of Gauss-Cantor sets and their applications to the study
of classical Markov spectrum\nby Polina Vytnova (University of Warwic
k) as part of One World Numeration seminar\n\n\nAbstract\nThe classical La
grange and Markov spectra are subsets of the real line which arise in conn
ection with some problems in theory Diophantine approximation theory. In 1
921 O. Perron gave a definition in terms of continued fractions\, which al
lowed to study the Markov and Lagrange spectra using limit sets of iterate
d function schemes. \n\nIn 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-Canto
r sets. \n\nThe talk is based on a joint work with C. Matheus\, C. G. More
ira and M. Pollicott.\n
LOCATION:https://researchseminars.org/talk/OWNS/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niclas Technau (University of Wisconsin - Madison)
DTSTART;VALUE=DATE-TIME:20210706T123000Z
DTEND;VALUE=DATE-TIME:20210706T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/52
DESCRIPTION:Title: Li
ttlewood and Duffin-Schaeffer-type problems in diophantine approximation\nby Niclas Technau (University of Wisconsin - Madison) as part of One W
orld Numeration seminar\n\n\nAbstract\nGallagher's theorem describes the m
ultiplicative diophantine approximation rate of a typical vector. Recently
Sam Chow and I establish a fully-inhomogeneous version of Gallagher's the
orem\, and a diophantine fibre refinement. In this talk I outline the proo
f\, and the tools involved in it.\n
LOCATION:https://researchseminars.org/talk/OWNS/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Oleg Karpenkov (University of Liverpool)
DTSTART;VALUE=DATE-TIME:20210907T123000Z
DTEND;VALUE=DATE-TIME:20210907T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/53
DESCRIPTION:Title: On
Hermite's problem\, Jacobi-Perron type algorithms\, and Dirichlet groups<
/a>\nby Oleg Karpenkov (University of Liverpool) as part of One World Nume
ration seminar\n\n\nAbstract\nIn this talk we introduce a new modification
of the Jacobi-Perron algorithm in the three dimensional case. This algori
thm is periodic for the case of totally-real conjugate cubic vectors. To t
he best of our knowledge this is the first Jacobi-Perron type algorithm fo
r which the cubic periodicity is proven. This provides an answer in the to
tally-real case to the question of algebraic periodicity for cubic irratio
nalities posed in 1848 by Ch.Hermite.\n\nWe will briefly discuss a new app
roach 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 alg
ebraic irrationalities.\n
LOCATION:https://researchseminars.org/talk/OWNS/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Henna Koivusalo (University of Vienna)
DTSTART;VALUE=DATE-TIME:20200602T123000Z
DTEND;VALUE=DATE-TIME:20200602T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/54
DESCRIPTION:Title: Li
near repetition in polytopal cut and project sets\nby Henna Koivusalo
(University of Vienna) as part of One World Numeration seminar\n\n\nAbstra
ct\nCut and project sets are aperiodic point patterns obtained by projecti
ng an irrational slice of the integer lattice to a subspace. One way of cl
assifying aperiodic sets is to study repetition of finite patterns\, where
sets with linear pattern repetition can be considered as the most ordered
aperiodic sets. \nRepetitivity of a cut and project set depends on the sl
ope and shape of the irrational slice. The cross-section of the slice is k
nown as the window. In an earlier work it was shown that for cut and proje
ct 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 r
esult for other polytopal windows\, under mild geometric conditions. A key
step in the proof is a decomposition of the cut and project scheme\, whic
h allows us to make sense of condition (ii) for general polytopal windows.
\n
LOCATION:https://researchseminars.org/talk/OWNS/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steve Jackson (University of North Texas)
DTSTART;VALUE=DATE-TIME:20210914T123000Z
DTEND;VALUE=DATE-TIME:20210914T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/55
DESCRIPTION:Title: De
scriptive complexity in numeration systems\nby Steve Jackson (Universi
ty of North Texas) as part of One World Numeration seminar\n\n\nAbstract\n
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\, $\\beta$-expansion
s\, and GLS expansions. In work with Mance and Vandehey we show that the n
umbers which are continued fraction normal but not base $b$ normal is comp
lete 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) onl
y known previously assuming the generalized Riemann hypothesis.\n
LOCATION:https://researchseminars.org/talk/OWNS/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Maria Siskaki (University of Illinois at Urbana-Champaign)
DTSTART;VALUE=DATE-TIME:20210921T123000Z
DTEND;VALUE=DATE-TIME:20210921T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/56
DESCRIPTION:Title: Th
e distribution of reduced quadratic irrationals arising from continued fra
ction expansions\nby Maria Siskaki (University of Illinois at Urbana-C
hampaign) as part of One World Numeration seminar\n\n\nAbstract\nIt is kno
wn that the reduced quadratic irrationals arising from regular continued f
raction 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\, an
d then by Boca\, that gives the error in the asymptotic behavior of this d
istribution. Moreover\, I will present the respective result for the distr
ibution of reduced quadratic irrationals that arise from even (joint work
with F. Boca) and odd continued fractions.\n
LOCATION:https://researchseminars.org/talk/OWNS/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philipp Hieronymi (Universität Bonn)
DTSTART;VALUE=DATE-TIME:20210928T123000Z
DTEND;VALUE=DATE-TIME:20210928T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/57
DESCRIPTION:Title: A
strong version of Cobham's theorem\nby Philipp Hieronymi (Universität
Bonn) as part of One World Numeration seminar\n\n\nAbstract\nLet $k\,l>1$
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 r
ecognized by some finite automaton. Cobham’s famous theorem states that
a subset of the natural numbers is both $k$-recognizable and $l$-recogniza
ble 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-defi
nable. 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 th
e first-order logical theory of $(\\mathbb{N}\,+\,X)$ is decidable. Our wo
rk strengthens and depends on earlier work of Villemaire and Bès.\n\nThe
essence of Cobham's theorem is that recognizability depends strongly on th
e choice of the base $k$. Our results strengthens this: two non-Presburger
definable sets that are recognizable in multiplicatively independent base
s\, are not only distinct\, but together computationally intractable over
Presburger arithmetic.\n\nThis is joint work with Christian Schulz.\n
LOCATION:https://researchseminars.org/talk/OWNS/57/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lulu Fang (Nanjing University of Science and Technology)
DTSTART;VALUE=DATE-TIME:20211005T123000Z
DTEND;VALUE=DATE-TIME:20211005T130000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/58
DESCRIPTION:Title: On
upper and lower fast Khintchine spectra in continued fractions\nby Lu
lu Fang (Nanjing University of Science and Technology) as part of One Worl
d Numeration seminar\n\n\nAbstract\nLet $\\psi:\\mathbb{N}\\to \\mathbb{R}
^+$ be a function satisfying $\\psi(n)/n\\to \\infty$ as $n \\to \\infty$.
\nWe investigate from a multifractal analysis point of view the growth sp
eed of the sums $\\sum^n_{k=1}\\log a_k(x)$ \nwith respect to $\\psi(n)$\,
where $x=[a_1(x)\,a_2(x)\,\\cdots]$ denotes the continued fraction expans
ion of $x\\in (0\,1)$. \nThe (upper\, lower) fast Khintchine spectrum is d
efined as the Hausdorff dimension of the set of points $x\\in(0\,1)$ \nfor
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 \nhave been studied by Fan\,
Liao \,Wang \\& Wu (2013\, 2016)\, Liao \\& Rams (2016). In this talk\, w
e will give a new look \nat the fast Khintchine spectrum\, and provide a f
ull description of upper and lower fast Khintchine spectra. The latter \ni
mproves a result of Liao and Rams (2016).\n
LOCATION:https://researchseminars.org/talk/OWNS/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liangang Ma (Binzhou University)
DTSTART;VALUE=DATE-TIME:20211012T123000Z
DTEND;VALUE=DATE-TIME:20211012T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/59
DESCRIPTION:Title: In
flection points in the Lyapunov spectrum for IFS on intervals\nby Lian
gang Ma (Binzhou University) as part of One World Numeration seminar\n\nAb
stract: TBA\n
LOCATION:https://researchseminars.org/talk/OWNS/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Taylor Jones (University of North Texas)
DTSTART;VALUE=DATE-TIME:20211005T130000Z
DTEND;VALUE=DATE-TIME:20211005T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/60
DESCRIPTION:Title: On
the Existence of Numbers with Matching Continued Fraction and Decimal Exp
ansion\nby Taylor Jones (University of North Texas) as part of One Wor
ld Numeration seminar\n\n\nAbstract\nA Trott number in base 10 is one whos
e continued fraction expansion agrees with its base 10 expansion in the se
nse that $[0\;a_1\,a_2\,\\dots] = 0.(a_1)(a_2) \\cdots$ where $(a_i)$ repr
esents the string of digits of $a_i$. As an example $[0\;3\,29\,54\,7\,\\d
ots] = 0.329547\\cdots$.\nAn 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 wh
ich $b$? We discuss the history of the problem\, and give a heuristic proc
ess for constructing such numbers. We show that $T_{10}$ is indeed non-emp
ty\, and uncountable. With more delicate techniques\, a complete classific
ation may be given to all $b$ for which $T_b$ is non-empty. We also discus
s some further results\, such as a (non-trivial) upper bound on the Hausdo
rff dimension of $T_b$\, as well as the question of whether the intersecti
on of $T_b$ and $T_c$ can be non-empty.\n
LOCATION:https://researchseminars.org/talk/OWNS/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mélodie Lapointe (Université de Paris)
DTSTART;VALUE=DATE-TIME:20211019T123000Z
DTEND;VALUE=DATE-TIME:20211019T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/61
DESCRIPTION:Title: q-
analog of the Markoff injectivity conjecture\nby Mélodie Lapointe (Un
iversité de Paris) as part of One World Numeration seminar\n\n\nAbstract\
nThe Markoff injectivity conjecture states that $w\\mapsto\\mu(w)_{12}$ is
injective on the set of Christoffel words where $\\mu:\\{\\mathtt{0}\,\\m
athtt{1}\\}^*\\to\\mathrm{SL}_2(\\mathbb{Z})$ is a certain homomorphism an
d $M_{12}$ is the entry above the diagonal of a $2\\times2$ matrix $M$. Re
cently\, Leclere and Morier-Genoud (2021) proposed a $q$-analog $\\mu_q$ o
f $\\mu$ such that $\\mu_{q\\to1}(w)_{12}=\\mu(w)_{12}$ is the Markoff num
ber associated to the Christoffel word $w$. We show that there exists an o
rder $<_{radix}$ 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 i
ndeterminate $q$ with nonnegative integer coefficients. Therefore\, for ev
ery $q>0$\, the map $\\{\\mathtt{0}\,\\mathtt{1}\\}^*\\to\\mathbb{R}$ defi
ned by $w\\mapsto\\mu_q(w)_{12}$ is increasing thus injective over the lan
guage of a balanced sequence. The proof uses an equivalence between bala
nced sequences satisfying some Markoff property and indistinguishable asym
ptotic pairs.\n
LOCATION:https://researchseminars.org/talk/OWNS/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Baake (Universität Bielefeld)
DTSTART;VALUE=DATE-TIME:20211026T123000Z
DTEND;VALUE=DATE-TIME:20211026T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/62
DESCRIPTION:Title: Sp
ectral aspects of aperiodic dynamical systems\nby Michael Baake (Unive
rsität Bielefeld) as part of One World Numeration seminar\n\n\nAbstract\n
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 th
e diffraction measure can be used for this purpose. First\, we consider so
me concequences of inflation rules on the spectra via renormalisation\, an
d how to use it to exclude absolutely continuous componenta. Second\, we t
ake a look at a class of dynamical systems of number-theoretic origin\, ho
w they fit into the spectral picture\, and what (other) methods there are
to distinguish them.\n
LOCATION:https://researchseminars.org/talk/OWNS/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pieter Allaart (University of North Texas)
DTSTART;VALUE=DATE-TIME:20211102T133000Z
DTEND;VALUE=DATE-TIME:20211102T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/63
DESCRIPTION:Title: On
the existence of Trott numbers relative to multiple bases\nby Pieter
Allaart (University of North Texas) as part of One World Numeration semina
r\n\n\nAbstract\nTrott numbers are real numbers in the interval $(0\,1)$ w
hose continued fraction expansion equals their base-$b$ expansion\, in a c
ertain liberal but natural sense. They exist in some bases\, but not in al
l. 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 proper
ties 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 certai
nly "no"\, a full proof of this seems currently out of reach. But we obtai
n some interesting partial answers by using a deep theorem from Diophantin
e approximation.\n
LOCATION:https://researchseminars.org/talk/OWNS/63/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Zhiqiang Wang (East China Normal University)
DTSTART;VALUE=DATE-TIME:20211109T133000Z
DTEND;VALUE=DATE-TIME:20211109T140000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/64
DESCRIPTION:Title: Ho
w inhomogeneous Cantor sets can pass a point\nby Zhiqiang Wang (East C
hina Normal University) as part of One World Numeration seminar\n\n\nAbstr
act\nAbstract: 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$.\n
LOCATION:https://researchseminars.org/talk/OWNS/64/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lucía Rossi (Montanuniversität Leoben)
DTSTART;VALUE=DATE-TIME:20211116T133000Z
DTEND;VALUE=DATE-TIME:20211116T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/65
DESCRIPTION:Title: Ra
tional self-affine tiles associated to (nonstandard) digit systems\nby
Lucía Rossi (Montanuniversität Leoben) as part of One World Numeration
seminar\n\n\nAbstract\nIn this talk we will introduce the notion of ration
al self-affine tiles\, which are fractal-like sets that arise as the solut
ion 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 vect
ors. 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 tile
s for the case where the base is a rational matrix with irreducible charac
teristic polynomial. We present some tiling results that generalize the on
es obtained by Lagarias and Wang: we consider arbitrary expanding rational
matrices as bases\, and simultaneously allow the digit sets to be nonstan
dard (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.\
n
LOCATION:https://researchseminars.org/talk/OWNS/65/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sascha Troscheit (Universität Wien)
DTSTART;VALUE=DATE-TIME:20211123T133000Z
DTEND;VALUE=DATE-TIME:20211123T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/66
DESCRIPTION:Title: An
alogues of Khintchine's theorem for random attractors\nby Sascha Trosc
heit (Universität Wien) as part of One World Numeration seminar\n\n\nAbst
ract\nKhintchine’s theorem is an important result in number theory which
links the Lebesgue measure of certain limsup sets with the convergence/di
vergence of naturally occurring volume sums. This behaviour has been obser
ved for deterministic fractal sets and inspired by this we investigate the
random settings. Introducing randomisation into the problem makes some pa
rts more tractable\, while posing separate new challenges. In this talk\,
I will present joint work with Simon Baker where we provide sufficient con
ditions for a large class of stochastically self-similar and self-affine a
ttractors to have positive Lebesgue measure.\n
LOCATION:https://researchseminars.org/talk/OWNS/66/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jamie Walton (University of Glasgow)
DTSTART;VALUE=DATE-TIME:20211207T133000Z
DTEND;VALUE=DATE-TIME:20211207T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/67
DESCRIPTION:Title: Ex
tending the theory of symbolic substitutions to compact alphabets\nby
Jamie Walton (University of Glasgow) as part of One World Numeration semin
ar\n\n\nAbstract\nIn 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 topo
logy for which the substitution is continuous. Such substitutions have bee
n considered before\, in particular by Durand\, Ormes and Petite for zero-
dimensional alphabets\, and Queffélec in the constant length case. We fin
d a simple condition which ensures that an associated substitution operato
r is quasi-compact\, which we conjecture to always be satisfied for primit
ive substitutions on countable alphabets. In the primitive case this impli
es the existence of a unique natural tile length function and\, for a reco
gnisable substitution\, that the associated shift space is uniquely ergodi
c. The main tools come from the theory of positive operators on Banach spa
ces. Very few prerequisites will be assumed\, and the theory will be demon
strated via examples.\n
LOCATION:https://researchseminars.org/talk/OWNS/67/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Younès Tierce (Université de Rouen Normandie)
DTSTART;VALUE=DATE-TIME:20211109T140000Z
DTEND;VALUE=DATE-TIME:20211109T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/68
DESCRIPTION:Title: Ex
tensions of the random beta-transformation\nby Younès Tierce (Univers
ité de Rouen Normandie) as part of One World Numeration seminar\n\n\nAbst
ract\nLet $\\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" extens
ion of the random $\\beta$-transformation\, providing a new proof of the e
xistence and unicity of an absolutely continuous invariant probability mea
sure\, and an expression of the density of this measure. This extension sh
ows off some nice renewal times\, and we use these to prove that the natur
al extension of the system is a Bernoulli automorphism.\n
LOCATION:https://researchseminars.org/talk/OWNS/68/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Valérie Berthé\, Pierre Arnoux\, ...
DTSTART;VALUE=DATE-TIME:20211214T130000Z
DTEND;VALUE=DATE-TIME:20211214T150000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/69
DESCRIPTION:Title: Sp
ecial session commemorating Shunji Ito (1943-2021)\nby Valérie Berth
é\, Pierre Arnoux\, ... as part of One World Numeration seminar\n\n\nAbst
ract\nIntroduction by Pierre Arnoux\, short talk by Valérie Berthé\, con
tributions by Maki Furukado\, Cor Kraaikamp\, Hui Rao\, Robbie Robinson\,
Shin'Ichi Yasutomi\, Shigeki Akiyama\, and Hiromi Ei.\n
LOCATION:https://researchseminars.org/talk/OWNS/69/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Fan Lü (Sichuan Normal University)
DTSTART;VALUE=DATE-TIME:20211221T133000Z
DTEND;VALUE=DATE-TIME:20211221T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/70
DESCRIPTION:Title: Mu
ltiplicative Diophantine approximation in the parameter space of beta-dyna
mical system\nby Fan Lü (Sichuan Normal University) as part of One Wo
rld Numeration seminar\n\n\nAbstract\nBeta-transformation is a special kin
d 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 poin
t in $(0\,1]$\, we consider the set of parameters $(\\alpha\, \\beta)$\, s
uch that the multiple $\\|T^n_{\\alpha}(x)\\|\\|T^n_{\\beta}(x)\\|$ is wel
l approximated or badly approximated. The Gallagher-type question\, Jarní
k-type question as well as the badly approximable pairs\, i.e.\, Littlewoo
d-type question are studied in detail.\n
LOCATION:https://researchseminars.org/talk/OWNS/70/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Philipp Gohlke (Universität Bielefeld)
DTSTART;VALUE=DATE-TIME:20220111T133000Z
DTEND;VALUE=DATE-TIME:20220111T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/71
DESCRIPTION:Title: Ze
ro measure spectrum for multi-frequency Schrödinger operators\nby Phi
lipp Gohlke (Universität Bielefeld) as part of One World Numeration semin
ar\n\n\nAbstract\nCantor spectrum of zero Lebesgue measure is a striking f
eature of Schrödinger operators associated with certain models of aperiod
ic order\, like primitive substitution systems or Sturmian subshifts. This
is known to follow from a condition introduced by Boshernitzan that estab
lishes 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 almo
st every translation admits a natural coding such that the associated subs
hift satisfies the Boshernitzan criterion (joint work with J.Chaika\, D.Da
manik and J.Fillman).\n
LOCATION:https://researchseminars.org/talk/OWNS/71/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Agamemnon Zafeiropoulos (NTNU)
DTSTART;VALUE=DATE-TIME:20220118T133000Z
DTEND;VALUE=DATE-TIME:20220118T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/72
DESCRIPTION:Title: Th
e order of magnitude of Sudler products\nby Agamemnon Zafeiropoulos (N
TNU) as part of One World Numeration seminar\n\n\nAbstract\nGiven an irrat
ional $\\alpha \\in [0\,1] \\smallsetminus \\mathbb{Q}$\, we define the co
rresponding 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 in
teger b\, the following hold: \n\n- If $b\\leq 5$\, then $\\liminf_{N\\to
\\infty}P_N(\\alpha) >0$ and $\\limsup_{N\\to \\infty} P_N(\\alpha)/N < \\
infty$. \n\n-If $b\\geq 6$\, then $\\liminf_{N\\to \\infty}P_N(\\alpha) =
0$ and $\\limsup_{N\\to \\infty} P_N(\\alpha)/N = \\infty$. \n\nWe also pr
esent an analogue of the previous result for arbitrary quadratic irrationa
ls (joint work with S. Grepstad and M. Neumueller).\n
LOCATION:https://researchseminars.org/talk/OWNS/72/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claudio Bonanno (Università di Pisa)
DTSTART;VALUE=DATE-TIME:20220125T133000Z
DTEND;VALUE=DATE-TIME:20220125T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/73
DESCRIPTION:Title: In
finite ergodic theory and a tree of rational pairs\nby Claudio Bonanno
(Università di Pisa) as part of One World Numeration seminar\n\n\nAbstra
ct\nThe study of the continued fraction expansions of real numbers by ergo
dic methods is now a classical and well-known part of the theory of dynami
cal systems. Less is known for the multi-dimensional expansions. I will pr
esent an ergodic approach to a two-dimensional continued fraction algorith
m introduced by T. Garrity\, and show how to get a complete tree of ration
al pairs by using the Farey sum of fractions. The talk is based on joint w
ork with A. Del Vigna and S. Munday.\n
LOCATION:https://researchseminars.org/talk/OWNS/73/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Magdaléna Tinková (Czech Technical University in Prague)
DTSTART;VALUE=DATE-TIME:20220208T133000Z
DTEND;VALUE=DATE-TIME:20220208T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/74
DESCRIPTION:Title: Un
iversal quadratic forms\, small norms and traces in families of number fie
lds\nby Magdaléna Tinková (Czech Technical University in Prague) as
part of One World Numeration seminar\n\n\nAbstract\nIn 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 Vít
ězslav Kala.\n
LOCATION:https://researchseminars.org/talk/OWNS/74/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Coons (Universität Bielefeld)
DTSTART;VALUE=DATE-TIME:20220308T133000Z
DTEND;VALUE=DATE-TIME:20220308T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/75
DESCRIPTION:Title: A
spectral theory of regular sequences\nby Michael Coons (Universität B
ielefeld) as part of One World Numeration seminar\n\n\nAbstract\nA few yea
rs ago\, Michael Baake and I introduced a probability measure associated t
o Stern’s diatomic sequence\, an example of a regular sequence—sequenc
es which generalise constant length substitutions to infinite alphabets. I
n this talk\, I will discuss extensions of these results to more general r
egular sequences as well as further properties of these measures. This is
joint work with several people\, including Michael Baake\, James Evans\, Z
achary Groth and Neil Manibo.\n
LOCATION:https://researchseminars.org/talk/OWNS/75/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonas Jankauskas (Vilnius University)
DTSTART;VALUE=DATE-TIME:20220201T133000Z
DTEND;VALUE=DATE-TIME:20220201T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/76
DESCRIPTION:Title: Di
git systems with rational base matrix over lattices\nby Jonas Jankausk
as (Vilnius University) as part of One World Numeration seminar\n\n\nAbstr
act\nLet $A$ be a matrix with rational entries and no eigenvalue in absolu
te value smaller than 1. Let $\\mathbb{Z}^d[A]$ be the minimal $A$-invaria
nt $\\mathbb{Z}$-module\, generated by integer vectors and the matrix $A$.
In 2018\, we have shown that one can find a finite set $D$ of vectors\, s
uch that each element of $\\mathbb{Z}^d[A]$ has a finite radix expansion i
n 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\, t
hat enable us to build finite digit systems over lattices using reasonably
small digit sets\, and even to do some practical computations with them o
n a computer. Tools that we use are the generalized rotation bases with di
git sets that have 'good' convex properties\, the semi-direct ('twisted')
sums of such rotational digit systems\, and the special\, 'restricted' ver
sion 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. Thus
waldner\, "Rational Matrix Digit Systems"\, to appear in "Linear and Multi
linear Algebra".\n
LOCATION:https://researchseminars.org/talk/OWNS/76/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wolfgang Steiner (CNRS\, Université de Paris)
DTSTART;VALUE=DATE-TIME:20220215T133000Z
DTEND;VALUE=DATE-TIME:20220215T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/77
DESCRIPTION:Title: Un
ique double base expansions\nby Wolfgang Steiner (CNRS\, Université d
e Paris) as part of One World Numeration seminar\n\n\nAbstract\nFor pairs
of real bases $\\beta_0\, \\beta_1 > 1$\, we study expansions of the form\
n$\\sum_{k=1}^\\infty i_k / (\\beta_{i_1} \\beta_{i_2} \\cdots \\beta_{i_k
})$\nwith digits $i_k \\in \\{0\,1\\}$.\nWe characterise the pairs admitti
ng non-trivial unique expansions as well as those admitting uncountably ma
ny unique expansions\, extending recent results of Neunhäuserer (2021) an
d Zou\, Komornik and Lu (2021).\nSimilarly to the study of unique $\\beta$
-expansions with three digits by the speaker (2020)\, this boils down to d
etermining the cardinality of binary shifts defined by lexicographic inequ
alities.\nLabarca and Moreira (2006) characterised when such a shift is em
pty\, at most countable or uncountable\, depending on the position of the
lower and upper bounds with respect to Thue-Morse-Sturmian words. \n\nThis
is joint work with Vilmos Komornik and Yuru Zou.\n
LOCATION:https://researchseminars.org/talk/OWNS/77/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Krenn (Universität Salzburg)
DTSTART;VALUE=DATE-TIME:20220301T133000Z
DTEND;VALUE=DATE-TIME:20220301T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/78
DESCRIPTION:Title: $k
$-regular sequences: Asymptotics and Decidability\nby Daniel Krenn (Un
iversität Salzburg) as part of One World Numeration seminar\n\n\nAbstract
\nA 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 w
e compute it? If not\, is it just a lack of methods or are the underlying
decision questions recursively solvable (i.e.\, decidable in a computation
al sense)? We will discuss answers to these questions. To round off the pi
cture\, we will consider further decidability questions around $k$-regular
sequences and the subclass of $k$-automatic sequences.\n\nThis is based o
n joint works with Clemens Heuberger and with Jeffrey Shallit.\n
LOCATION:https://researchseminars.org/talk/OWNS/78/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Pierre Popoli (Université de Lorraine)
DTSTART;VALUE=DATE-TIME:20220315T133000Z
DTEND;VALUE=DATE-TIME:20220315T143000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/79
DESCRIPTION:Title: Ma
ximum order complexity for some automatic and morphic sequences along poly
nomial values\nby Pierre Popoli (Université de Lorraine) as part of O
ne World Numeration seminar\n\n\nAbstract\nAutomatic sequences are not sui
table sequences for cryptographic applications since both their subword co
mplexity and their expansion complexity are small\, and their correlation
measure of order 2 is large. These sequences are highly predictable despit
e having a large maximum order complexity. However\, recent results show t
hat polynomial subsequences of automatic sequences\, such as the Thue-Mors
e sequence or the Rudin-Shapiro sequence\, are better candidates for pseud
orandom sequences. A natural generalization of automatic sequences are mor
phic sequences\, given by a fixed point of a prolongeable morphism that is
not necessarily uniform. In this talk\, I will present my results on lowe
rs bounds for the maximum order complexity of the Thue-Morse sequence\, th
e Rudin-Shapiro sequence and the sum of digits function in Zeckendorf base
\, which are respectively automatics and morphic sequences.\n
LOCATION:https://researchseminars.org/talk/OWNS/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tingyu Zhang (East China Normal University)
DTSTART;VALUE=DATE-TIME:20220329T123000Z
DTEND;VALUE=DATE-TIME:20220329T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/80
DESCRIPTION:Title: Ra
ndom $\\beta$-transformation on fat Sierpiński gasket\nby Tingyu Zhan
g (East China Normal University) as part of One World Numeration seminar\n
\n\nAbstract\nWe define the notions of greedy\, lazy and random transforma
tions on fat Sierpiński gasket. We determine the bases\, for which the sy
stem has a unique measure of maximal entropy and an invariant measure of p
roduct type\, with one coordinate being absolutely continuous with respect
to Lebesgue measure. \n\nThis is joint work with K. Dajani and W. Li.\n
LOCATION:https://researchseminars.org/talk/OWNS/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jungwon Lee (University of Warwick)
DTSTART;VALUE=DATE-TIME:20220405T123000Z
DTEND;VALUE=DATE-TIME:20220405T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/81
DESCRIPTION:Title: Dy
namics of Ostrowski skew-product: Limit laws and Hausdorff dimensions\
nby Jungwon Lee (University of Warwick) as part of One World Numeration se
minar\n\n\nAbstract\nWe discuss a dynamical study of the Ostrowski skew-pr
oduct map in the context of inhomogeneous Diophantine approximation. We pl
an to outline the setup/ strategy based on transfer operator analysis and
applications in arithmetic of number fields (joint with Valérie Berthé).
\n
LOCATION:https://researchseminars.org/talk/OWNS/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eda Cesaratto (Univ. Nac. de Gral. Sarmiento & CONICET)
DTSTART;VALUE=DATE-TIME:20220412T123000Z
DTEND;VALUE=DATE-TIME:20220412T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/82
DESCRIPTION:Title: Lo
chs-type theorems beyond positive entropy\nby Eda Cesaratto (Univ. Nac
. de Gral. Sarmiento & CONICET) as part of One World Numeration seminar\n\
n\nAbstract\nLochs' theorem and its generalizations are conversion theorem
s that relate the number of digits determined in one expansion of a real n
umber 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 res
ults holding almost everywhere\, or in measure\, involving the entropy. Th
is is the viewpoint we develop here. In order to deal with sequences of pa
rtitions 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 s
tate 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. \n\nThis is joint work with Valérie Berthé (IR
IF)\, Pablo Rotondo (U. Gustave Eiffel) and Martín Safe (Univ. Nac. del S
ur & CONICET\, Argentina).\n
LOCATION:https://researchseminars.org/talk/OWNS/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paulina Cecchi Bernales (Universidad de Chile)
DTSTART;VALUE=DATE-TIME:20220419T123000Z
DTEND;VALUE=DATE-TIME:20220419T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/83
DESCRIPTION:Title: Co
boundaries and eigenvalues of finitary S-adic systems\nby Paulina Cecc
hi Bernales (Universidad de Chile) as part of One World Numeration seminar
\n\n\nAbstract\nAn S-adic system is a shift space obtained by performing a
n 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 eige
nvalues of substitution shifts\, and where coboundaries appear as a key to
ol. We will introduce the notion of S-adic coboundaries and present some r
esults 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. \n\nThis is joint work with Valérie Berthé and
Reem Yassawi.\n
LOCATION:https://researchseminars.org/talk/OWNS/83/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolas Chevallier (Université de Haute Alsace)
DTSTART;VALUE=DATE-TIME:20220503T123000Z
DTEND;VALUE=DATE-TIME:20220503T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/84
DESCRIPTION:Title: Be
st Diophantine approximations in the complex plane with Gaussian integers<
/a>\nby Nicolas Chevallier (Université de Haute Alsace) as part of One Wo
rld Numeration seminar\n\n\nAbstract\nStarting with the minimal vectors in
lattices over Gaussian integers in $\\C^2$\, we define a algorithm that f
inds the sequence of minimal vectors of any unimodular lattice in $\\C^2$.
\nRestricted to lattices associated with complex numbers this algorithm fi
nd all the best Diophantine approximations of a complex numbers.\nFollowin
g Doeblin\, Lenstra\, Bosma\, Jager and Wiedijk\, we study the limit distr
ibution of the sequence of products $(u_{n1}u_{n2})_n$ where $(u _n=( u_{n
1}\,u_{n2} ))_n$ 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 distribut
ion of the sequence of products of almost all unimodular lattices.\n
LOCATION:https://researchseminars.org/talk/OWNS/84/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Vilmos Komornik (Shenzhen University and Université de Strasbourg
)
DTSTART;VALUE=DATE-TIME:20220517T123000Z
DTEND;VALUE=DATE-TIME:20220517T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/85
DESCRIPTION:Title: To
pology of univoque sets in real base expansions\nby Vilmos Komornik (S
henzhen University and Université de Strasbourg) as part of One World Num
eration seminar\n\n\nAbstract\nWe report on a recent joint paper with Mart
ijn de Vries and Paola Loreti. Given a positive integer $M$ and a real num
ber $1 < q\\le M+1$\, 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 $x=\\sum_{i=1}^{\\infty}c_iq^{-i}$. G
eneralizing many earlier results\, we investigate the topological properti
es 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.\n
LOCATION:https://researchseminars.org/talk/OWNS/85/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Émilie Charlier (Université de Liège)
DTSTART;VALUE=DATE-TIME:20220524T123000Z
DTEND;VALUE=DATE-TIME:20220524T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/86
DESCRIPTION:Title: Sp
ectrum\, algebraicity and normalization in alternate bases\nby Émilie
Charlier (Université de Liège) as part of One World Numeration seminar\
n\n\nAbstract\nThe first aim of this work is to give information about the
algebraic properties of alternate bases determining sofic systems. We exh
ibit two conditions: one necessary and one sufficient. Comparing the setti
ng 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 resu
lt concerning normalization of real bases representations. Under some suit
able condition (i.e.\, our previous sufficient condition for being a sofic
system)\, we prove that the normalization function is computable by a fin
ite Büchi automaton\, and furthermore\, we effectively construct such an
automaton. An important tool in our study is the spectrum of numeration sy
stems associated with alternate bases. For our purposes\, we use a general
ized concept of spectrum associated with a complex base and complex digits
\, and we study its topological properties. \n\nThis is joint work with C
élia Cisternino\, Zuzana Masáková and Edita Pelantová.\n
LOCATION:https://researchseminars.org/talk/OWNS/86/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Verónica Becher (Universidad de Buenos Aires & CONICET Argentina)
DTSTART;VALUE=DATE-TIME:20220531T123000Z
DTEND;VALUE=DATE-TIME:20220531T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/87
DESCRIPTION:Title: Po
isson generic real numbers\nby Verónica Becher (Universidad de Buenos
Aires & CONICET Argentina) as part of One World Numeration seminar\n\n\nA
bstract\nYears ago Zeev Rudnick defined the Poisson generic real numbers a
s those where the number of occurrences of the long strings in the initia
l segments of their fractional expansions in some base have the Poisson di
stribution. Yuval Peres and Benjamin Weiss proved that almost all real num
bers\, with respect to Lebesgue measure\, are Poisson generic. They also s
howed that Poisson genericity implies Borel normality but the two notions
do not coincide\, witnessed by the famous Champernowne constant. We rec
ently showed that there are computable Poisson generic real numbers and th
at all Martin-Löf real numbers are Poisson generic. \nThis is joint work
Nicolás Álvarez and Martín Mereb.\n
LOCATION:https://researchseminars.org/talk/OWNS/87/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sophie Morier-Genoud (Université Reims Champagne Ardenne)
DTSTART;VALUE=DATE-TIME:20220607T123000Z
DTEND;VALUE=DATE-TIME:20220607T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/88
DESCRIPTION:Title: q-
analogues of real numbers\nby Sophie Morier-Genoud (Université Reims
Champagne Ardenne) as part of One World Numeration seminar\n\n\nAbstract\n
Classical sequences of numbers often lead to interesting q-analogues. The
most popular among them are certainly the q-integers and the q-binomial co
efficients 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 defini
tion of q-rationals naturally extends the one of q-integers and leads to a
ratio of polynomials with positive integer coefficients. I will explain t
he 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 furt
her developments of the theory\, leading to the notion of q-irrationals an
d q-unimodular matrices.\n
LOCATION:https://researchseminars.org/talk/OWNS/88/
END:VEVENT
BEGIN:VEVENT
SUMMARY:James A. Yorke (University of Maryland)
DTSTART;VALUE=DATE-TIME:20220621T123000Z
DTEND;VALUE=DATE-TIME:20220621T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/89
DESCRIPTION:Title: La
rge and Small Chaos Models\nby James A. Yorke (University of Maryland)
as part of One World Numeration seminar\n\n\nAbstract\nTo set the scene\,
I will discuss one large model\, a whole-Earth model for predicting the w
eather\, and how to initialize such a model and what aspects of chaos are
essential. Then I will discuss a couple related “very simple” maps tha
t tell us a great deal about very complex models. The results on simple mo
dels are new. I will discuss the logistic map mx(1-x). Its dynamics can ma
ke us rethink climate models. Also\, we have created a piecewise linear ma
p on a 3D cube that is unstable in 2 dimensions in some places and unstabl
e in 1 in others. It has a dense set of periodic points that are 1 D unsta
ble 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?\n
LOCATION:https://researchseminars.org/talk/OWNS/89/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Charlene Kalle (Universiteit Leiden)
DTSTART;VALUE=DATE-TIME:20220705T123000Z
DTEND;VALUE=DATE-TIME:20220705T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/90
DESCRIPTION:Title: Ra
ndom Lüroth expansions\nby Charlene Kalle (Universiteit Leiden) as pa
rt of One World Numeration seminar\n\n\nAbstract\nSince the introduction o
f Lüroth expansions by Lüroth in his paper from 1883 many results have a
ppeared on their approximation properties. In 1990 Kalpazidou\, Knopfmache
r and Knopfmacher introduced alternating Lüroth expansions and studied th
eir properties. A comparison between the two and other comparable number s
ystems was then given by Barrionuevo\, Burton\, Dajani and Kraaikamp in 19
96. In this talk we introduce a family of random dynamical systems that pr
oduce many Lüroth type expansions at once. Topics that we consider are pe
riodic expansions\, universal expansions\, speed of convergence and approx
imation coefficients. This talk is based on joint work with Marta Maggioni
.\n
LOCATION:https://researchseminars.org/talk/OWNS/90/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ruofan Li (South China University of Technology)
DTSTART;VALUE=DATE-TIME:20220712T123000Z
DTEND;VALUE=DATE-TIME:20220712T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/91
DESCRIPTION:Title: Ra
tional numbers in $\\times b$-invariant sets\nby Ruofan Li (South Chin
a University of Technology) as part of One World Numeration seminar\n\n\nA
bstract\nLet $b \\ge 2$ be an integer and $S$ be a finite non-empty set of
primes not containing divisors of $b$. For any $\\times b$-invariant\, no
n-dense subset $A$ of $[0\,1)$\, we prove the finiteness of rational numbe
rs in $A$ whose denominators can only be divided by primes in $S$. A quant
itative result on the largest prime divisors of the denominators of ration
al numbers in $A$ is also obtained. \nThis is joint work with Bing Li and
Yufeng Wu.\n
LOCATION:https://researchseminars.org/talk/OWNS/91/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benedict Sewell (Alfréd Rényi Institute)
DTSTART;VALUE=DATE-TIME:20220913T123000Z
DTEND;VALUE=DATE-TIME:20220913T133000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/92
DESCRIPTION:Title: An
upper bound on the box-counting dimension of the Rauzy gasket\nby Ben
edict Sewell (Alfréd Rényi Institute) as part of One World Numeration se
minar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/OWNS/92/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Niels Langeveld (Montanuniversität Leoben)
DTSTART;VALUE=DATE-TIME:20220927T120000Z
DTEND;VALUE=DATE-TIME:20220927T130000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/93
DESCRIPTION:Title: $N
$-continued fractions and $S$-adic sequences\nby Niels Langeveld (Mont
anuniversität Leoben) as part of One World Numeration seminar\n\n\nAbstra
ct\nGiven the $N$-continued fraction of a number $x$\, we construct $N$-co
ntinued fraction sequences in the same spirit as Sturmian sequences can be
constructed from regular continued fractions. These sequences are infinit
e words over a two letter alphabet obtained as the limit of a directive se
quence of certain substitutions (they are S-adic sequences). By viewing th
em as a generalisation of Sturmian sequences it is natural to study balanc
edness. 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 algorith
m. This talk is joint work with Lucía Rossi and Jörg Thuswaldner.\n
LOCATION:https://researchseminars.org/talk/OWNS/93/
END:VEVENT
BEGIN:VEVENT
SUMMARY:David Siukaev (Higher School of Economics)
DTSTART;VALUE=DATE-TIME:20221004T120000Z
DTEND;VALUE=DATE-TIME:20221004T123000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/94
DESCRIPTION:Title: Ex
actness and ergodicity of certain Markovian multidimensional fraction algo
rithms\nby David Siukaev (Higher School of Economics) as part of One W
orld Numeration seminar\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/OWNS/94/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lukas Spiegelhofer (Montanuniversität Leoben)
DTSTART;VALUE=DATE-TIME:20221011T120000Z
DTEND;VALUE=DATE-TIME:20221011T130000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/95
DESCRIPTION:Title: Pr
imes as sums of Fibonacci numbers\nby Lukas Spiegelhofer (Montanuniver
sität Leoben) as part of One World Numeration seminar\n\n\nAbstract\nWe p
rove that the Zeckendorf sum-of-digits function of prime numbers\, $z(p)$\
, is uniformly distributed in residue classes.\nThe main ingredient that m
ade this proof possible is the study of very sparse arithmetic subsequence
s of $z(n)$. In other words\, we will meet the level of distribution.\nOur
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 G
owers norm related to $z(n)$.\nOur method of proof yields examples of subs
titutive sequences that are orthogonal to the Möbius function (cf. Sarnak
's conjecture).\n\nThis is joint work with Michael Drmota and Clemens Mül
lner (TU Wien).\n
LOCATION:https://researchseminars.org/talk/OWNS/95/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandra Skripchenko (Higher School of Economics)
DTSTART;VALUE=DATE-TIME:20221004T123000Z
DTEND;VALUE=DATE-TIME:20221004T130000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/96
DESCRIPTION:Title: Br
uin-Troubetzkoy family of interval translation mappings: a new glance\
nby Alexandra Skripchenko (Higher School of Economics) as part of One Worl
d Numeration seminar\n\n\nAbstract\nIn 2002 H. Bruin and S. Troubetzkoy de
scribed a special class of interval translation mappings on three interval
s. They showed that in this class the typical ITM could be reduced to an i
nterval exchange transformations. They also proved that generic ITM of the
ir class that can not be reduced to IET is uniquely ergodic. \n\nWe sugges
t an alternative proof of the first statement and get a stronger version o
f the second one. It is a joint work in progress with Mauro Artigiani and
Pascal Hubert.\n
LOCATION:https://researchseminars.org/talk/OWNS/96/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Álvaro Bustos-Gajardo (The Open University)
DTSTART;VALUE=DATE-TIME:20221025T120000Z
DTEND;VALUE=DATE-TIME:20221025T130000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/97
DESCRIPTION:Title: Qu
asi-recognizability and continuous eigenvalues of torsion-free S-adic syst
ems\nby Álvaro Bustos-Gajardo (The Open University) as part of One Wo
rld Numeration seminar\n\n\nAbstract\nWe discuss combinatorial and dynamic
al descriptions of S-adic systems generated by sequences of constant-lengt
h morphisms between alphabets of bounded size. For this purpose\, we intro
duce the notion of quasi-recognisability\, a strictly weaker version of re
cognisability but which is indeed enough to reconstruct several classical
arguments of the theory of constant-length substitutions in this more gene
ral context. Furthermore\, we identify a large family of directive sequenc
es\, which we call "torsion-free"\, for which quasi-recognisability is obt
ained naturally\, and can be improved to actual recognisability with relat
ive ease.\n\nUsing these notions we give S-adic analogues of the notions o
f column number and height for substitutions\, including dynamical and com
binatorial interpretations of each\, and give a general characterisation o
f the maximal equicontinuous factor of the identified family of S-adic shi
fts\, showing as a consequence that in this context all continuous eigenva
lues must be rational. As well\, we employ the tools developed for a first
approach to the measurable case.\n\nThis is a joint work with Neil Mañib
o and Reem Yassawi.\n
LOCATION:https://researchseminars.org/talk/OWNS/97/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wen Wu (South China University of Technology)
DTSTART;VALUE=DATE-TIME:20221108T130000Z
DTEND;VALUE=DATE-TIME:20221108T140000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/98
DESCRIPTION:Title: Fr
om the Thue-Morse sequence to the apwenian sequences\nby Wen Wu (South
China University of Technology) as part of One World Numeration seminar\n
\n\nAbstract\nIn this talk\, we will introduce a class of $\\pm 1$ sequenc
es\, called the apwenian sequences. The Hankel determinants of these $\\p
m1$ sequences share the same property as the Hankel determinants of the Th
ue-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 qu
estions are also discussed for $0$-$1$ apwenian sequences. This talk is b
ased on joint work with Y.-J. Guo and G.-N. Han.\n
LOCATION:https://researchseminars.org/talk/OWNS/98/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Faustin Adiceam (Université Paris-Est Créteil)
DTSTART;VALUE=DATE-TIME:20221122T130000Z
DTEND;VALUE=DATE-TIME:20221122T140000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/99
DESCRIPTION:Title: Ba
dly approximable vectors and Littlewood-type problems\nby Faustin Adic
eam (Université Paris-Est Créteil) as part of One World Numeration semin
ar\n\n\nAbstract\nBadly approximable vectors are fractal sets enjoying ric
h Diophantine properties. In this respect\, they play a crucial role in ma
ny problems well beyond Number Theory and Fractal Geometry (e.g.\, in sign
al processing\, in mathematical physics and in convex geometry). \n\nAfter
outlining some of the latest developments in this very active area of res
earch\, we will take an interest in the Littlewood conjecture (c. 1930) an
d in its variants which all admit a natural formulation in terms of proper
ties satisfied by badly approximable vectors. We will then show how ideas
emerging from the mathematical theory of quasicrystals\, from numeration s
ystems and from the theory of aperiodic tilings have recently been used to
refute the so-called t-adic Littlewood conjecture. \n\nAll necessary conc
epts will be defined in the talk. Joint with Fred Lunnon (Maynooth) and Er
ez Nesharim (Technion\, Haifa).\n
LOCATION:https://researchseminars.org/talk/OWNS/99/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yufei Chen (TU Delft)
DTSTART;VALUE=DATE-TIME:20221018T120000Z
DTEND;VALUE=DATE-TIME:20221018T130000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/100
DESCRIPTION:Title: M
atching of orbits of certain $N$-expansions with a finite set of digits\nby Yufei Chen (TU Delft) as part of One World Numeration seminar\n\n\nA
bstract\nIn 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 t
he underlying Gauss-map $T_{\\alpha}$ is equal. In this paper we show that
for all $N\\in\n$\, $N\\geq 2$\, such plateaux exist. In order to show th
at the entropy is constant on such plateaux\, we obtain the underlying pla
nar natural extension of the maps $T_{\\alpha}$\, the $T_{\\alpha}$-invari
ant 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 pr
operty called matching.\n
LOCATION:https://researchseminars.org/talk/OWNS/100/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Seul Bee Lee (Institute for Basic Science)
DTSTART;VALUE=DATE-TIME:20221115T130000Z
DTEND;VALUE=DATE-TIME:20221115T140000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/101
DESCRIPTION:Title: R
egularity properties of Brjuno functions associated with by-excess\, odd a
nd even continued fractions\nby Seul Bee Lee (Institute for Basic Scie
nce) as part of One World Numeration seminar\n\n\nAbstract\nAn 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 nu
mbers comes from the study of one variable analytic small divisor problems
. In 1988\, J.-C. Yoccoz introduced the Brjuno function which characterize
s the Brjuno numbers to estimate the size of Siegel disks. In this talk\,
we introduce Brjuno-type functions associated with by-excess\, odd and eve
n continued fractions with a number theoretical motivation. Then we discus
s the $L^p$ and the Hölder regularity properties of the difference betwee
n the classical Brjuno function and the Brjuno-type functions. This is joi
nt work with Stefano Marmi.\n
LOCATION:https://researchseminars.org/talk/OWNS/101/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manuel Hauke (TU Graz)
DTSTART;VALUE=DATE-TIME:20221129T130000Z
DTEND;VALUE=DATE-TIME:20221129T140000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/102
DESCRIPTION:Title: T
he asymptotic behaviour of Sudler products\nby Manuel Hauke (TU Graz)
as part of One World Numeration seminar\n\n\nAbstract\nGiven an irrational
number $\\alpha$\, we study the asymptotic behaviour of the Sudler produc
t defined by $P_N(\\alpha) = \\prod_{r=1}^N 2 \\lvert \\sin \\pi r \\alpha
\\rvert$\, which appears in many different areas of mathematics.\nIn this
talk\, we explain the connection between the size of $P_N(\\alpha)$ and t
he Ostrowski expansion of $N$ with respect to $\\alpha$.\nWe 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 t
he continued fraction expansion of $\\alpha$ exceeds $7$ infinitely often\
, and show that the value $7$ is optimal.\n\nFor Lebesgue-almost every $\\
alpha$\, we can prove more: we show that for every non-decreasing function
$\\psi: (0\,\\infty) \\to (0\,\\infty)$ with $\\sum_{k=1}^{\\infty} \\fra
c{1}{\\psi(k)} = \\infty$ and\n$\\liminf_{k \\to \\infty} \\psi(k)/(k \\lo
g k)$ sufficiently large\, the conditions $\\log P_N(\\alpha) \\leq -\\psi
(\\log N)$\, $\\log P_N(\\alpha) \\geq \\psi(\\log N)$ hold on sets of upp
er density $1$ respectively $1/2$.\n
LOCATION:https://researchseminars.org/talk/OWNS/102/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christoph Bandt (Universität Greifswald)
DTSTART;VALUE=DATE-TIME:20221206T130000Z
DTEND;VALUE=DATE-TIME:20221206T140000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/103
DESCRIPTION:Title: A
utomata generated topological spaces and self-affine tilings\nby Chris
toph Bandt (Universität Greifswald) as part of One World Numeration semin
ar\n\n\nAbstract\nNumeration assigns symbolic sequences as addresses to po
ints in a space X. There are points which get multiple addresses. It is k
nown that these identifications describe the topology of X and can often b
e determined by an automaton. Here we define a corresponding class of auto
mata 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 sim
ple class of examples.\n
LOCATION:https://researchseminars.org/talk/OWNS/103/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hiroki Takahasi (Keio University)
DTSTART;VALUE=DATE-TIME:20221213T130000Z
DTEND;VALUE=DATE-TIME:20221213T140000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/104
DESCRIPTION:Title: D
istribution of cycles for one-dimensional random dynamical systems\nby
Hiroki Takahasi (Keio University) as part of One World Numeration seminar
\n\n\nAbstract\nWe consider an independently identically distributed rando
m dynamical system generated by finitely many\, non-uniformly expanding Ma
rkov interval maps with a finite number of branches.\nAssuming a topologic
ally mixing condition and the uniqueness of equilibrium state for the asso
ciated 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 t
opologically mixing Axiom A diffeomorphisms. \n\nThis talk is based on the
preprint arXiv:2108.05522. If time permits\, I will mention some future p
erspectives in this project.\n
LOCATION:https://researchseminars.org/talk/OWNS/104/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roswitha Hofer (JKU Linz)
DTSTART;VALUE=DATE-TIME:20230110T130000Z
DTEND;VALUE=DATE-TIME:20230110T140000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/105
DESCRIPTION:Title: E
xact order of discrepancy of normal numbers\nby Roswitha Hofer (JKU Li
nz) as part of One World Numeration seminar\n\n\nAbstract\nIn the talk we
discuss some previous results on the discrepancy of normal numbers and con
sider 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 $b$.
\n\nSo far the best upper bounds for $D_N$ for explicitly known normal nu
mbers in base $2$ are of the form $ND_N\\ll\\log^2 N$. The first example i
s 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 Gerhar
d Larcher that guarantees $ND_N\\gg \\log^2 N$ for Levin's binary normal n
umber. 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 norma
l number with a better growth of $ND_N$ in $N$. The recent result for Levi
n'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.\n
LOCATION:https://researchseminars.org/talk/OWNS/105/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Slade Sanderson (Utrecht University)
DTSTART;VALUE=DATE-TIME:20230131T130000Z
DTEND;VALUE=DATE-TIME:20230131T140000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/106
DESCRIPTION:Title: M
atching for parameterised symmetric golden maps\nby Slade Sanderson (U
trecht University) as part of One World Numeration seminar\n\n\nAbstract\n
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 ‘symm
etric 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 char
acterize the matching phenomenon in our setting\, present explicit invaria
nt measures and frequencies of digits of signed $\\beta$-expansions\, and-
--time permitting---show further implications for a family of piecewise li
near maps which arise as jump transformations of the symmetric golden maps
. \n\nJoint with Karma Dajani.\n
LOCATION:https://researchseminars.org/talk/OWNS/106/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kiko Kawamura (University of North Texas)
DTSTART;VALUE=DATE-TIME:20230124T130000Z
DTEND;VALUE=DATE-TIME:20230124T140000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/107
DESCRIPTION:Title: T
he partial derivative of Okamoto's functions with respect to the parameter
\nby Kiko Kawamura (University of North Texas) as part of One World Nu
meration seminar\n\n\nAbstract\nOkamoto's functions were introduced in 200
5 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 par
ameter\, one can produce interesting examples: Perkins' nowhere differenti
able function\, Bourbaki-Katsuura function and Cantor's Devil's staircase
function. \n\nIn this talk\, we consider the partial derivative of Okomoto
's functions with respect to the parameter $a$. We place a significant foc
us on $a = 1/3$ to describe the properties of a nowhere differentiable fun
ction $K(x)$ for which the set of points of infinite derivative produces a
n example of a measure zero set with Hausdorff dimension 1.\n\nThis is a j
oint work with T. Mathis and M.Paizanis (undergraduate students) and N.Dal
aklis (graduate student). The talk is very accessible and includes many co
mputer graphics.\n
LOCATION:https://researchseminars.org/talk/OWNS/107/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Derong Kong (Chongqing University)
DTSTART;VALUE=DATE-TIME:20230307T130000Z
DTEND;VALUE=DATE-TIME:20230307T140000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/108
DESCRIPTION:Title: C
ritical values for the beta-transformation with a hole at 0\nby Derong
Kong (Chongqing University) as part of One World Numeration seminar\n\n\n
Abstract\nGiven $\\beta \\in (1\,2]$\, let $T$ be the $\\beta$-transformat
ion on the unit circle $[0\,1)$. For $t \\in [0\,1)$ let $K(t)$ be the sur
vivor 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 di
mension 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 whe
n $t$ is passing through this critical value. In this paper we will descri
be this critical value and analyze its interesting properties. Our strateg
y to find the critical value depends on certain substitutions of Farey wor
ds and a renormalization scheme from dynamical systems. This is joint work
with Pieter Allaart.\n
LOCATION:https://researchseminars.org/talk/OWNS/108/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Demi Allen (University of Exeter)
DTSTART;VALUE=DATE-TIME:20230321T130000Z
DTEND;VALUE=DATE-TIME:20230321T140000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/109
DESCRIPTION:Title: D
iophantine Approximation for systems of linear forms - some comments on in
homogeneity\, monotonicity\, and primitivity\nby Demi Allen (Universit
y of Exeter) as part of One World Numeration seminar\n\n\nAbstract\nDiopha
ntine Approximation is a branch of Number Theory in which the central them
e 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 spec
ified 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 Diopha
ntine Approximation. In this talk I will discuss the generalisation of Khi
ntchine's Theorem to the setting of approximation for systems of linear fo
rms. I will focus mainly on the topic of inhomogeneous approximation for s
ystems of linear forms. Time permitting\, I may also discuss approximation
for systems of linear forms subject to certain primitivity constraints. T
his talk will be based on joint work with Felipe Ramirez (Wesleyan\, US).\
n
LOCATION:https://researchseminars.org/talk/OWNS/109/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ale Jan Homburg (University of Amsterdam\, VU University Amsterdam
)
DTSTART;VALUE=DATE-TIME:20230207T130000Z
DTEND;VALUE=DATE-TIME:20230207T140000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/110
DESCRIPTION:Title: I
terated function systems of linear expanding and contracting maps on the u
nit interval\nby Ale Jan Homburg (University of Amsterdam\, VU Univers
ity Amsterdam) as part of One World Numeration seminar\n\n\nAbstract\nWe a
nalyze the two-point motions of iterated function systems on the unit inte
rval generated by expanding and contracting affine maps\, where the expans
ion and contraction rates are determined by a pair $(M\,N)$ of integers.\n
\nThis dynamics depends on the Lyapunov exponent.\n\nFor a negative Lyapun
ov exponent we establish synchronization\, meaning convergence of orbits w
ith different initial points. For a vanishing Lyapunov exponent we establi
sh intermittency\, where orbits are close for a set of iterates of full de
nsity\, 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.\n\nFor nonnegative Lyapun
ov exponent and pairs $(M\,N)$ that are multiplicatively dependent integer
s\, we provide explicit expressions for absolutely continuous stationary m
easures of the two-point dynamics. These stationary measures are infinite
$\\sigma$-finite measures in the case of zero Lyapunov exponent.\n\nThis i
s joint work with Charlene Kalle.\n
LOCATION:https://researchseminars.org/talk/OWNS/110/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yining Hu (Huazhong University of Science and Technology)
DTSTART;VALUE=DATE-TIME:20230214T130000Z
DTEND;VALUE=DATE-TIME:20230214T140000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/111
DESCRIPTION:Title: A
lgebraic automatic continued fractions in characteristic 2\nby Yining
Hu (Huazhong University of Science and Technology) as part of One World Nu
meration seminar\n\n\nAbstract\nWe present two families of automatic seque
nces 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 famil
y $\\mathcal{G}$. The family $\\mathcal{G}$ contains all the iterated sum
s of sequences from the $\\mathcal{P}$ and more.\n
LOCATION:https://researchseminars.org/talk/OWNS/111/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Roland Zweimüller (University of Vienna)
DTSTART;VALUE=DATE-TIME:20230328T120000Z
DTEND;VALUE=DATE-TIME:20230328T130000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/112
DESCRIPTION:Title: V
ariations on a theme of Doeblin\nby Roland Zweimüller (University of
Vienna) as part of One World Numeration seminar\n\n\nAbstract\nStarting fr
om Doeblin's observation on the Poissonian nature of occurrences of large
digits in typical continued fraction expansions\, I will outline some rece
nt work on rare events in measure preserving systems (including spatiotemp
oral and local limit theorems) which\, in particular\, allows us to refine
Doeblin's statement in several ways. \n\n(Part of this is joint work with
Max Auer.)\n
LOCATION:https://researchseminars.org/talk/OWNS/112/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Anton Lukyanenko (George Mason University)
DTSTART;VALUE=DATE-TIME:20230418T120000Z
DTEND;VALUE=DATE-TIME:20230418T130000Z
DTSTAMP;VALUE=DATE-TIME:20230330T200748Z
UID:OWNS/113
DESCRIPTION:Title: S
erendipitous decompositions of higher-dimensional continued fractions\
nby Anton Lukyanenko (George Mason University) as part of One World Numera
tion seminar\n\n\nAbstract\nComplex continued fractions (CFs) represent a
complex number using a descending fraction with Gaussian integer coefficie
nts. The associated dynamical system is exact (Nakada 1981) with a piecewi
se-analytic invariant measure (Hensley 2006). Certain higher-dimensional C
Fs\, including CFs over quaternions\, octonions\, as well as the non-commu
tative Heisenberg group can be understood in a unified way using the Iwasa
wa CF framework (L-Vandehey 2022). Under some natural and robust assumptio
ns\, ergodicity of the associated systems can then be derived from a conne
ction to hyperbolic geodesic flow\, but stronger mixing results and inform
ation about the invariant measure remain elusive. Here\, we study Iwasawa
CFs under a more delicate serendipity assumption that yields the finite ra
nge condition\, allowing us to extend the Nakada-Hensley results to certai
n Iwasawa CFs over the quaternions\, octonions\, and in $\\mathbb{R}^3$.\n
\nThis is joint work with Joseph Vandehey.\n
LOCATION:https://researchseminars.org/talk/OWNS/113/
END:VEVENT
END:VCALENDAR