# 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`

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Tuesday, February 1, 2022, 14:30 CET (UTC +1)

Jonas Jankauskas (Vilniaus universitetas): Digit systems with rational base matrix over lattices (paper)

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

Tuesday, February 8, 2022, 14:30 CET (UTC +1)

Magdaléna Tinková (České vysoké učení technické v Praze): Universal quadratic forms, small norms and traces in families of number fields (paper)

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

Tuesday, February 15, 2022, 14:30 CET (UTC +1)

Wolfgang Steiner (CNRS, Université de Paris): Unique double base expansions

Tuesday, March 1, 2022, 14:30 CET (UTC +1)

Daniel Krenn (Universität Salzburg): TBA

Tuesday, March 8, 2022, 14:30 CET (UTC +1)

Michael Coons (Universität Bielefeld): TBA

## Past talks

January 25, 2022

Claudio Bonanno (Università di Pisa): Infinite ergodic theory and a tree of rational pairs (video) (slides) (paper)

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

January 18, 2022

Agamemnon Zafeiropoulos (Norges teknisk-naturvitenskapelige universitet): The order of magnitude of Sudler products (video) (slides) (paper1) (paper2)

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

- If b ≤ 5, then lim inf_{N→∞} P_{N}(α) > 0 and lim sup_{N→∞} P_{N}(α)/N < ∞.

- If b ≥ 6, then lim inf_{N→∞} P_{N}(α) = 0 and lim sup_{N→∞} P_{N}(α)/N = ∞.

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

January 11, 2022

Philipp Gohlke (Universität Bielefeld): Zero measure spectrum for multi-frequency Schrödinger operators (video)
(slides) (paper)

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

December 21, 2021

Fan Lü (Sichuan Normal University): Multiplicative Diophantine approximation in the parameter space of beta-dynamical system (video) (slides)

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

December 14, 2021

Special session commemorating Shunji Ito (1943-2021)

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

December 7, 2021

Jamie Walton (University of Nottingham): Extending the theory of symbolic substitutions to compact alphabets (video) (notes)

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

November 23, 2021

Sascha Troscheit (Universität Wien): Analogues of Khintchine's theorem for random attractors (video) (slides) (paper)

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

November 16, 2021

Lucía Rossi (Montanuniversität Leoben): Rational self-affine tiles associated to (nonstandard) digit systems (video) (slides) (paper)

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

November 9, 2021

Zhiqiang Wang (East China Normal University): How inhomogeneous Cantor sets can pass a point (video) (slides)

For x>0, we define Y(x) = { (a,b) : x ∈ E_{a,b}, a>0, b>0, a+b≤1 }, where the set E_{a,b} is the unique nonempty compact invariant set generated by the inhomogeneous IFS { f_{0}(x) = ax, f_1(x) = b(x+1) }. We show the set Y(x) is a Lebesgue null set with full Hausdorff dimension in R^{2}, and the intersection of sets Y(x_{1}), Y(x_{2}), ..., Y(x_{k}) still has full Hausdorff dimension in R^{2} for any finitely many positive real numbers x_{1}, x_{2}, ..., x_{k}.

November 9, 2021

Younès Tierce (Université de Rouen Normandie): Extensions of the random beta-transformation (video) (slides)

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

November 2, 2021

Pieter Allaart (University of North Texas): On the existence of Trott numbers relative to multiple bases (video) (slides) (paper)

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

October 26, 2021

Michael Baake (Universität Bielefeld): Spectral aspects of aperiodic dynamical systems (slides)

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

October 19, 2021

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

The Markoff injectivity conjecture states that the map w→μ(w)_{12} is injective on the set of Christoffel words where μ: {0,1}*→SL_{2}(Z) is a certain homomorphism and M_{12} is the entry above the diagonal of a 2x2 matrix M. Recently, Leclere and Morier-Genoud (2021) proposed a q-analog μ_{q} of μ such that μ_{q→1}(w)_{12} = μ(w)_{12} is the Markoff number associated to the Christoffel word w. We show that there exists an order <_{radix} on {0,1}* such that for every balanced sequence s ∈ {0,1}^{Z} and for all factors u,v in the language of s with u <_{radix} v, the difference μ_{q}(v)_{12} - μ_{q}(u)_{12} is a nonzero polynomial of indeterminate q with nonnegative integer coefficients. Therefore, for every q>0, the map {0,1}*→R defined by w→μ(w)_{12} is increasing thus injective over the language of a balanced sequence. The proof uses an equivalence between balanced sequences satisfying some Markoff property and indistinguishable asymptotic pairs.

October 12, 2021

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

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.

October 5, 2021

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

October 5, 2021

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

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).

September 28, 2021

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

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.

September 21, 2021

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.

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.