## One world numeration seminar

#### Jour, heure et lieu

Le mardi à 14h30, online

Le calendrier des séances (format iCal).

Pour ajouter le calendrier des séances à votre agenda favori, souscrire au calendrier en indiquant ce lien.

#### Contact(s)

This is an international online seminar on numeration systems and related topics. If you want to participate, please write to numeration@irif.fr. For more information, in particular slides and videos of past talks, visit Numeration - OWNS homepage.

### Prochaines séances

One world numeration seminar

Mardi 1 février 2022, 14 heures 30, Online

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

One world numeration seminar

Mardi 8 février 2022, 14 heures 30, Online

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

One world numeration seminar

Mardi 15 février 2022, 14 heures 30, Online

**Wolfgang Steiner** (IRIF) *Unique double base expansions*

### Séances passées

#### Année 2022

One world numeration seminar

Mardi 25 janvier 2022, 14 heures 30, Online

**Claudio Bonanno** (Università di Pisa) *Infinite ergodic theory and a tree of rational pairs*

One world numeration seminar

Mardi 18 janvier 2022, 14 heures 30, Online

**Agamemnon Zafeiropoulos** (NTNU) *The order of magnitude of Sudler products*

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

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

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

One world numeration seminar

Mardi 11 janvier 2022, 14 heures 30, Online

**Philipp Gohlke** (Universität Bielefeld) *Zero measure spectrum for multi-frequency Schrödinger operators*

#### Année 2021

One world numeration seminar

Mardi 21 décembre 2021, 14 heures 30, Online

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

One world numeration seminar

Mardi 7 décembre 2021, 14 heures 30, Online

**Jamie Walton** (University of Nottingham) *Extending the theory of symbolic substitutions to compact alphabets*

One world numeration seminar

Mardi 23 novembre 2021, 14 heures 30, Online

**Sascha Troscheit** (Universität Wien) *Analogues of Khintchine's theorem for random attractors*

One world numeration seminar

Mardi 16 novembre 2021, 14 heures 30, Online

**Lucía Rossi** (Montanuniversität Leoben) *Rational self-affine tiles associated to (nonstandard) digit systems*

One world numeration seminar

Mardi 9 novembre 2021, 14 heures 30, Online

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

One world numeration seminar

Mardi 9 novembre 2021, 15 heures, Online

**Younès Tierce** (Université de Rouen Normandie) *Extensions of the random beta-transformation*

One world numeration seminar

Mardi 2 novembre 2021, 14 heures 30, Online

**Pieter Allaart** (University of North Texas) *On the existence of Trott numbers relative to multiple bases*

One world numeration seminar

Mardi 26 octobre 2021, 14 heures 30, Online

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

One world numeration seminar

Mardi 19 octobre 2021, 14 heures 30, Online

**Mélodie Lapointe** (IRIF) *q-analog of the Markoff injectivity conjecture*

One world numeration seminar

Mardi 12 octobre 2021, 14 heures 30, Online

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

One world numeration seminar

Mardi 5 octobre 2021, 14 heures 30, Online

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

One world numeration seminar

Mardi 5 octobre 2021, 15 heures, Online

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

One world numeration seminar

Mardi 28 septembre 2021, 14 heures 30, Online

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

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.

One world numeration seminar

Mardi 21 septembre 2021, 14 heures 30, Online

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

One world numeration seminar

Mardi 14 septembre 2021, 14 heures 30, Online

**Steve Jackson** (University of North Texas) *Descriptive complexity in numeration systems*

One world numeration seminar

Mardi 7 septembre 2021, 14 heures 30, Online

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

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.

One world numeration seminar

Mardi 6 juillet 2021, 14 heures 30, Online

**Niclas Technau** (University of Wisconsin - Madison) *Littlewood and Duffin-Schaeffer-type problems in diophantine approximation*

One world numeration seminar

Mardi 29 juin 2021, 14 heures 30, Online

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

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.

One world numeration seminar

Mardi 22 juin 2021, 14 heures 30, Online

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

One world numeration seminar

Mardi 15 juin 2021, 14 heures 30, Online

**Sam Chow** (University of Warwick) *Dyadic approximation in the Cantor set*

One world numeration seminar

Mardi 8 juin 2021, 14 heures 30, Online

**Shigeki Akiyama** (University of Tsukuba) *Counting balanced words and related problems*

One world numeration seminar

Mardi 1 juin 2021, 14 heures 30, Online

**Bastián Espinoza** (Université de Picardie Jules Verne and Universidad de Chile) *Automorphisms and factors of finite topological rank systems*

One world numeration seminar

Mardi 25 mai 2021, 14 heures 30, Online

**Charles Fougeron** (IRIF) *Dynamics of simplicial systems and multidimensional continued fraction algorithms*

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.

One world numeration seminar

Mardi 18 mai 2021, 14 heures 30, Online

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

One world numeration seminar

Mardi 11 mai 2021, 14 heures 30, Online

**Giulio Tiozzo** (University of Toronto) *The bifurcation locus for numbers of bounded type*

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.

One world numeration seminar

Mardi 4 mai 2021, 16 heures, Online

**Tushar Das** (University of Wisconsin - La Crosse) *Hausdorff Hensley Good & Gauss*

One world numeration seminar

Mardi 27 avril 2021, 14 heures 30, Online

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

One world numeration seminar

Mardi 20 avril 2021, 14 heures 30, Online

**Ayreena Bakhtawar** (La Trobe University) *Metrical theory for the set of points associated with the generalized Jarnik-Besicovitch set*

One world numeration seminar

Mardi 13 avril 2021, 14 heures 30, Online

**Andrew Mitchell** (University of Birmingham) *Measure theoretic entropy of random substitutions*

One world numeration seminar

Mardi 30 mars 2021, 14 heures 30, Online

**Michael Drmota** (TU Wien) *(Logarithmic) Densities for Automatic Sequences along Primes and Squares*

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.

One world numeration seminar

Mardi 23 mars 2021, 14 heures 30, Online

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

One world numeration seminar

Mardi 16 mars 2021, 14 heures 30, Online

**Alexandra Skripchenko** (Higher School of Economics) *Double rotations and their ergodic properties*

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

One world numeration seminar

Mardi 9 mars 2021, 14 heures 30, Online

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

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

One world numeration seminar

Mardi 2 mars 2021, 16 heures, Online

**Vitaly Bergelson** (Ohio State University) *Normal sets in (ℕ,+) and (ℕ,×)*

One world numeration seminar

Mardi 23 février 2021, 14 heures 30, Online

**Seulbee Lee** (Scuola Normale Superiore di Pisa) *Odd-odd continued fraction algorithm*

One world numeration seminar

Mardi 16 février 2021, 14 heures 30, Online

**Gerardo González Robert** (Universidad Nacional Autónoma de México) *Good's Theorem for Hurwitz Continued Fractions*

One world numeration seminar

Mardi 9 février 2021, 14 heures 30, Online

**Clemens Müllner** (TU Wien) *Multiplicative automatic sequences*

One world numeration seminar

Mardi 2 février 2021, 14 heures 30, Online

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

One world numeration seminar

Mardi 26 janvier 2021, 14 heures 30, Online

**Carlo Carminati** (Università di Pisa) *Prevalence of matching for families of continued fraction algorithms: old and new results*

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.

One world numeration seminar

Mardi 19 janvier 2021, 14 heures 30, Online

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

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

In the case that beta is Pisot one can study the spectrum of beta dynamically using substitutions or cut and project schemes, and this allows one to see lots of local structure in the spectrum. There are higher dimensional analogues for other algebraic integers. In this talk we will define a random walk on the spectrum of beta and show how, with appropriate renormalisation, this leads to an infinite stationary measure on the spectrum. This measure has local structure analagous to that of the spectrum itself. Furthermore, this measure has deep links with the Bernoulli convolution, and in particular new criteria for the absolute continuity of Bernoulli convolutions can be stated in terms of the ergodic properties of these measures.

One world numeration seminar

Mardi 5 janvier 2021, 14 heures 30, Online

**Claire Merriman** (Ohio State University) *alpha-odd continued fractions*

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.

#### Année 2020

One world numeration seminar

Mardi 15 décembre 2020, 14 heures 30, Online

**Lukas Spiegelhofer** (Montanuniversität Leoben) *The digits of n+t*

(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).

One world numeration seminar

Mardi 8 décembre 2020, 14 heures 30, Online

**Tanja Isabelle Schindler** (Scuola Normale Superiore di Pisa) *Limit theorems on counting large continued fraction digits*

One world numeration seminar

Mardi 1 décembre 2020, 14 heures 30, Online

**Michael Barnsley** (Australian National University) *Rigid fractal tilings*

One world numeration seminar

Mardi 17 novembre 2020, 14 heures 30, Online

**Jacques Sakarovitch** (IRIF, CNRS et Télécom Paris) *The carry propagation of the successor function*

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).

One world numeration seminar

Mardi 10 novembre 2020, 14 heures 30, Online

**Pieter Allaart** (University of North Texas) *On the smallest base in which a number has a unique expansion*

One world numeration seminar

Mardi 3 novembre 2020, 14 heures 30, Online

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

One world numeration seminar

Mardi 27 octobre 2020, 14 heures 30, Online

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

One world numeration seminar

Mardi 20 octobre 2020, 14 heures 30, Online

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

One world numeration seminar

Mardi 13 octobre 2020, 14 heures 30, Online

**Kan Jiang** (Ningbo University) *Representations of real numbers on fractal sets*

One world numeration seminar

Mardi 6 octobre 2020, 14 heures 30, Online

**Francesco Veneziano** (University of Genova) *Finiteness and periodicity of continued fractions over quadratic number fields*

One world numeration seminar

Mardi 29 septembre 2020, 14 heures 30, Online

**Marta Maggioni** (Leiden University) *Random matching for random interval maps*

One world numeration seminar

Mardi 22 septembre 2020, 14 heures 30, Online

**Yotam Smilansky** (Rutgers University) *Multiscale Substitution Tilings*

One world numeration seminar

Mardi 15 septembre 2020, 14 heures 30, Online

**Jeffrey Shallit** (University of Waterloo) *Lazy Ostrowski Numeration and Sturmian Words*

One world numeration seminar

Mardi 8 septembre 2020, 14 heures 30, Online

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

One world numeration seminar

Mardi 1 septembre 2020, 14 heures 30, Online

**Bill Mance** (Adam Mickiewicz University in Poznań) *Hotspot Lemmas for Noncompact Spaces*

One world numeration seminar

Mardi 14 juillet 2020, 14 heures 30, Online

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

One world numeration seminar

Mardi 7 juillet 2020, 14 heures 30, Online

**Hajime Kaneko** (University of Tsukuba) *Analogy of Lagrange spectrum related to geometric progressions*

One world numeration seminar

Mardi 30 juin 2020, 14 heures 30, Online

**Niels Langeveld** (Leiden University) *Continued fractions with two non integer digits*

One world numeration seminar

Mardi 23 juin 2020, 14 heures 30, Online

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

One world numeration seminar

Mardi 16 juin 2020, 14 heures 30, Online

**Carlos Matheus** (CNRS, École Polytechnique) *Approximations of the Lagrange and Markov spectra*

One world numeration seminar

Mardi 9 juin 2020, 14 heures 30, Online

**Simon Baker** (University of Birmingham) *Equidistribution results for self-similar measures*

One world numeration seminar

Mardi 2 juin 2020, 14 heures 30, Online

**Henna Koivusalo** (University of Vienna) *Linear repetition in polytopal cut and project sets*

One world numeration seminar

Mardi 26 mai 2020, 14 heures 30, Online

**Célia Cisternino** (University of Liège) *Ergodic behavior of transformations associated with alternate base expansions*

One world numeration seminar

Mardi 19 mai 2020, 14 heures 30, Online

**Boris Solomyak** (University of Bar-Ilan) *On singular substitution Z-actions*

One world numeration seminar

Mardi 12 mai 2020, 14 heures 30, Online

**Olivier Carton** (Université de Paris) *Preservation of normality by selection*

One world numeration seminar

Mardi 5 mai 2020, 14 heures 30, Online

**Narad Rampersad** (University of Winnipeg) *Ostrowski numeration and repetitions in words*