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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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:Henna Koivusalo (University of Vienna)
DTSTART;VALUE=DATE-TIME:20200602T123000Z
DTEND;VALUE=DATE-TIME:20200602T133000Z
DTSTAMP;VALUE=DATE-TIME:20210419T083041Z
UID:OWNS/5
DESCRIPTION:Title: Lin
ear repetition in polytopal cut and project sets\nby Henna Koivusalo (
University of Vienna) as part of One World Numeration seminar\n\n\nAbstrac
t\nCut and project sets are aperiodic point patterns obtained by projectin
g an irrational slice of the integer lattice to a subspace. One way of cla
ssifying aperiodic sets is to study repetition of finite patterns\, where
sets with linear pattern repetition can be considered as the most ordered
aperiodic sets. \n\nRepetitivity of a cut and project set depends on the s
lope and shape of the irrational slice. The cross-section of the slice is
known as the window. In an earlier work it was shown that for cut and proj
ect sets with a cube window\, linear repetitivity holds if and only if the
following two conditions are satisfied: (i) the set has minimal complexit
y and (ii) the irrational slope satisfies a certain Diophantine condition.
In a new joint work with Jamie Walton\, we give a generalisation of this
result for other polytopal windows\, under mild geometric conditions. A ke
y step in the proof is a decomposition of the cut and project scheme\, whi
ch allows us to make sense of condition (ii) for general polytopal windows
.\n
LOCATION:https://researchseminars.org/talk/OWNS/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Simon Baker (University of Birmingham)
DTSTART;VALUE=DATE-TIME:20200609T123000Z
DTEND;VALUE=DATE-TIME:20200609T133000Z
DTSTAMP;VALUE=DATE-TIME:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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:20210419T083041Z
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
END:VCALENDAR