# One World Numeration Seminar

This is an online seminar on numeration systems and related topics (see the series of Numeration conferences), in the spirit of other One World Seminars; talks are on Zoom.
If you want to participate in the seminar, please contact the organisers (Shigeki Akiyama, Karma Dajani, Kevin Hare, Hajime Kaneko, Niels Langeveld, Lingmin Liao, Wolfgang Steiner) by email to `numeration@irif.fr`

.

Tuesday, September 21, 2021, 14:30 CEST (UTC +2)

Maria Siskaki (University of Illinois at Urbana-Champaign): The distribution of reduced quadratic irrationals arising from continued fraction expansions (video) (slides) (paper)

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

Tuesday, September 28, 2021, 14:30 CEST (UTC +2)

Philipp Hieronymi (Universität Bonn): A strong version of Cobham's theorem

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

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

This is joint work with Christian Schulz.

Tuesday, October 5, 2021, 14:30 CEST (UTC +2)

Lulu Fang (Nanjing University of Science and Technology): On upper and lower fast Khintchine spectra in continued fractions

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

Tuesday, October 5, 2021, 15:00 CEST (UTC +2)

Taylor Jones (University of North Texas): On the Existence of Numbers with Matching Continued Fraction and Decimal Expansion (paper)

A Trott number in base 10 is one whose continued fraction expansion agrees with its base 10 expansion in the sense that [0;a_{1},a_{2},...] = 0.(a_{1})(a_{2})... where (a_{i}) represents the string of digits of a_{i}. As an example [0;3,29,54,7,...] = 0.329547... An analogous definition may be given for a Trott number in any integer base b>1, the set of which we denote by T_{b}. The first natural question is whether T_{b} is empty, and if not, for which b? We discuss the history of the problem, and give a heuristic process for constructing such numbers. We show that T_{10} is indeed non-empty, and uncountable. With more delicate techniques, a complete classification may be given to all b for which T_{b} is non-empty. We also discuss some further results, such as a (non-trivial) upper bound on the Hausdorff dimension of T_{b}, as well as the question of whether the intersection of T_{b} and T_{c} can be non-empty.

Tuesday, October 12, 2021, 14:30 CEST (UTC +2)

Liangang Ma (Binzhou University): Inflection points in the Lyapunov spectrum for IFS on intervals

We plan to present the audience a general picture about regularity of the Lyapunov spectrum for some iterated function systems, with emphasis on its inflection points in case the spectrum is smooth. Some sharp or moderate relationship between the number of Lyapunov inflections and (essential) branch number of a linear system is clarified. As most numeration systems are non-linear ones, the corresponding relationship for these systems are still mysterious enough comparing with the linear systems.

Tuesday, October 19, 2021, 14:30 CEST (UTC +2)

Mélodie Lapointe (Université de Paris): q-analog of the Markoff injectivity conjecture (paper)

## Past talks

September 14, 2021

Steve Jackson (University of North Texas): Descriptive complexity in numeration systems (video) (slides) (paper1) (paper2)

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 Π_{3}^{0} complete set. In work with Airey, Kwietniak, and Mance we extend to other numerations systems such as continued fractions, 𝛽-expansions, and GLS expansions. In work with Mance and Vandehey we show that the numbers which are continued fraction normal but not base b normal is complete at the expected level of D_{2}(Π_{3}^{0}). An immediate corollary is that this set is uncountable, a result (due to Vandehey) only known previously assuming the generalized Riemann hypothesis.

September 7, 2021

Oleg Karpenkov (University of Liverpool): On Hermite's problem, Jacobi-Perron type algorithms, and Dirichlet groups (video)(slides) (paper)

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

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

July 6, 2021

Niclas Technau (University of Wisconsin - Madison): Littlewood and Duffin-Schaeffer-type problems in diophantine approximation (video) (slides) (paper)

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

June 29, 2021

Polina Vytnova (University of Warwick): Hausdorff dimension of Gauss-Cantor sets and their applications to the study of classical Markov spectrum (video) (slides) (paper)

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

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

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

June 22, 2021

Lingmin Liao (Université Paris-Est Créteil Val de Marne): Simultaneous Diophantine approximation of the orbits of the dynamical systems x2 and x3 (video) (slides)

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

June 15, 2021

Sam Chow (University of Warwick): Dyadic approximation in the Cantor set (video) (slides) (paper)

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

June 8, 2021

Shigeki Akiyama (University of Tsukuba): Counting balanced words and related problems (video) (slides) (paper)

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

June 1, 2021

Bastián Espinoza (Université de Picardie Jules Verne and Universidad de Chile): Automorphisms and factors of finite topological rank systems (video) (slides) (paper1) (paper2)

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

May 25, 2021

Charles Fougeron (Université de Paris): Dynamics of simplicial systems and multidimensional continued fraction algorithms (video) (slides) (paper)

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

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

May 18, 2021

Joseph Vandehey (University of Texas at Tyler): Solved and unsolved problems in normal numbers (video) (slides)

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

May 11, 2021

Giulio Tiozzo (University of Toronto): The bifurcation locus for numbers of bounded type (video) (slides) (journal) (arXiv)

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

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

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

Joint with C. Carminati.

May 4, 2021

Tushar Das (University of Wisconsin - La Crosse): Hausdorff Hensley Good & Gauss (video) (slides) (paper1) (paper2)

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

April 27, 2021

Boris Adamczewski (CNRS, Université Claude Bernard Lyon 1): Expansions of numbers in multiplicatively independent bases: Furstenberg's conjecture and finite automata (video) (slides) (paper)

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

April 20, 2021

Ayreena Bakhtawar (La Trobe University): Metrical theory for the set of points associated with the generalized Jarnik-Besicovitch set (video) (slides) (journal) (arXiv)

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

Tuesday, April 13, 2021, 14:30 CEST (UTC +2)

Andrew Mitchell (University of Birmingham): Measure theoretic entropy of random substitutions (video) (slides)

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

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

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

March 30, 2021

Michael Drmota (TU Wien): (Logarithmic) Densities for Automatic Sequences along Primes and Squares (video) (slides) (paper)

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

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

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

This is a joint work with Boris Adamczewski and Clemens Müllner.

March 23, 2021

Godofredo Iommi (Pontificia Universidad Católica de Chile): Arithmetic averages and normality in continued fractions (video) (slides)

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

March 16, 2021

Alexandra Skripchenko (Higher School of Economics): Double rotations and their ergodic properties (video) (paper)

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

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

March 9, 2021

Natalie Priebe Frank (Vassar College): The flow view and infinite interval exchange transformation of a recognizable substitution (video) (paper)

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

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

March 2, 2021

Vitaly Bergelson (Ohio State University): Normal sets in (ℕ,+) and (ℕ,×) (video) (slides) (journal) (arXiv)

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

February 23, 2021

Seul Bee Lee (Scuola Normale Superiore di Pisa): Odd-odd continued fraction algorithm (video) (slides) (paper)

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

February 16, 2021

Gerardo González Robert (Universidad Nacional Autónoma de México): Good's Theorem for Hurwitz Continued Fractions (video) (slides) (journal) (arXiv)

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

February 9, 2021

Clemens Müllner (TU Wien): Multiplicative automatic sequences (video) (slides) (paper)

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

February 2, 2021

Samuel Petite (Université de Picardie Jules Verne): Interplay between finite topological rank minimal Cantor systems, S-adic subshifts and their complexity (video) (slides) (paper)

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

January 26, 2021

Carlo Carminati (Università di Pisa): Prevalence of matching for families of continued fraction algorithms: old and new results (video) (slides) (animation1) (animation2) (paper)

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

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

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

January 19, 2021

Tom Kempton (University of Manchester): Bernoulli Convolutions and Measures on the Spectra of Algebraic Integers (video) (slides)

Given an algebraic integer 𝛽 and alphabet A = {-1,0,1}, the spectrum of 𝛽 is the set

\Sigma(\beta) := \{ \sum_{i=1}^n a_i \beta^i : n \in \mathbb{N}, a_i \in A \}.

In the case that 𝛽 is Pisot one can study the spectrum of 𝛽 dynamically using substitutions or cut and project schemes, and this allows one to see lots of local structure in the spectrum. There are higher dimensional analogues for other algebraic integers.

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

January 5, 2021

Claire Merriman (Ohio State University): α-odd continued fractions (video) (slides) (journal) (arXiv)

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

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

December 15, 2020

Lukas Spiegelhofer (Montanuniversität Leoben): The digits of n+t (video) (slides) (paper)

We study the binary sum-of-digits function s_{2} under addition of a constant t.
For each integer k, we are interested in the asymptotic density δ(k,t) of integers t such that s_{2}(n+t) - s_{2}(n) = k.
In this talk, we consider the following two questions.

(1) Do we have c_{t} = δ(0,t) + δ(1,t) + ... > 1/2? This is a conjecture due to T. W. Cusick (2011).

(2) What does the probability distribution defined by k → δ(k,t) look like?

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

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

December 8, 2020

Tanja Isabelle Schindler (Scuola Normale Superiore di Pisa): Limit theorems on counting large continued fraction digits (video) (slides) (journal) (arXiv)

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

December 1, 2020

Michael Barnsley (Australian National University): Rigid fractal tilings (video) (slides) (paper)

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

November 17, 2020

Jacques Sakarovitch (Irif, CNRS and Télécom Paris): The carry propagation of the successor function (video) (slides) (paper)

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

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

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

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

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

November 10, 2020, 14:30

Pieter Allaart (University of North Texas): On the smallest base in which a number has a unique expansion (video) (slides) (paper)

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

November 3, 2020

Tomáš Vávra (University of Waterloo): Distinct unit generated number fields and finiteness in number systems (video) (slides)

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

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

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

October 27, 2020

Mélodie Andrieu (Aix-Marseille University): A Rauzy fractal unbounded in all directions of the plane (slides)

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

October 20, 2020

Paul Surer (University of Natural Resources and Life Sciences, Vienna): Representations for complex numbers with integer digits (video) (slides)

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

October 13, 2020

Kan Jiang (Ningbo University): Representations of real numbers on fractal sets (video) (slides)

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

October 6, 2020

Francesco Veneziano (University of Genova): Finiteness and periodicity of continued fractions over quadratic number fields (video) (slides) (paper)

We consider continued fractions with partial quotients in the ring of integers of a quadratic number field K; a particular example of these continued fractions is the β-continued fraction introduced by Bernat. We show that for any quadratic Perron number β, the β-continued fraction expansion of elements in Q(β) is either finite of eventually periodic. We also show that for certain four quadratic Perron numbers β, the β-continued fraction represents finitely all elements of the quadratic field Q(β), thus answering questions of Rosen and Bernat.

Based on a joint work with Zuzana Masáková and Tomáš Vávra.

September 29, 2020

Marta Maggioni (Leiden University): Random matching for random interval maps (video) (slides) (paper)

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

Based on a joint work with Karma Dajani and Charlene Kalle.

September 22, 2020

Yotam Smilansky (Rutgers University): Multiscale Substitution Tilings (video) (slides) (paper)

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

Based on joint work with Yaar Solomon.

September 15, 2020

Jeffrey Shallit (University of Waterloo): Lazy Ostrowski Numeration and Sturmian Words (video) (slides) (paper)

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

September 8, 2020

Bing Li (South China University of Technology): Some fractal problems in beta-expansions (video) (slides)

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

September 1, 2020

Bill Mance (Adam Mickiewicz University in Poznań): Hotspot Lemmas for Noncompact Spaces (video) (slides)

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

July 14, 2020

Attila Pethő (University of Debrecen): On diophantine properties of generalized number systems - finite and periodic representations (video) (slides)

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

July 7, 2020

Hajime Kaneko (University of Tsukuba): Analogy of Lagrange spectrum related to geometric progressions (video) (slides) (paper)

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

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

June 30, 2020

Niels Langeveld (Leiden University): Continued fractions with two non integer digits (video) (slides)

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

June 23, 2020

Derong Kong (Chongqing University): Univoque bases of real numbers: local dimension, Devil's staircase and isolated points (video) (slides) (paper)

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

June 16, 2020

Carlos Matheus (CNRS, École Polytechnique): Approximations of the Lagrange and Markov spectra (video) (slides) (journal) (arXiv)

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

June 9, 2020

Simon Baker (University of Birmingham): Equidistribution results for self-similar measures (video) (slides) (paper)

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

June 2, 2020

Henna Koivusalo (University of Vienna): Linear repetition in polytopal cut and project sets (video) (slides)

Cut and project sets are aperiodic point patterns obtained by projecting an irrational slice of the integer lattice to a subspace. One way of classifying aperiodic sets is to study repetition of finite patterns, where sets with linear pattern repetition can be considered as the most ordered aperiodic sets.

Repetitivity of a cut and project set depends on the slope and shape of the irrational slice. The cross-section of the slice is known as the window. In an earlier work it was shown that for cut and project sets with a cube window, linear repetitivity holds if and only if the following two conditions are satisfied: (i) the set has minimal complexity and (ii) the irrational slope satisfies a certain Diophantine condition. In a new joint work with Jamie Walton, we give a generalisation of this result for other polytopal windows, under mild geometric conditions. A key step in the proof is a decomposition of the cut and project scheme, which allows us to make sense of condition (ii) for general polytopal windows.

May 26, 2020

Célia Cisternino (University of Liège): Ergodic behavior of transformations associated with alternate base expansions (video) (slides)

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

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

May 19, 2020

Boris Solomyak (University of Bar-Ilan): On singular substitution Z-actions (video) (slides) (paper)

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

This is a joint work with A. Bufetov.

May 12, 2020

Olivier Carton (Université de Paris): Preservation of normality by selection (video) (slides)

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

May 5, 2020

Narad Rampersad (University of Winnipeg): Ostrowski numeration and repetitions in words (video) (slides)

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