## One world numeration seminar

The calendar of events (iCal format).

In order to add the event calendar to your favorite agenda, subscribe to the calendar by using this link.

#### 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.

### Next talks

One world numeration seminar

Tuesday May 11, 2021, 2:30PM, 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

Tuesday May 18, 2021, 2:30PM, Online

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

One world numeration seminar

Tuesday May 25, 2021, 2:30PM, 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

Tuesday June 1, 2021, 2:30PM, Online

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

### Previous talks

#### Year 2021

One world numeration seminar

Tuesday May 4, 2021, 4PM, Online

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

One world numeration seminar

Tuesday April 27, 2021, 2:30PM, 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

Tuesday April 20, 2021, 2:30PM, Online

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

One world numeration seminar

Tuesday April 13, 2021, 2:30PM, Online

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

One world numeration seminar

Tuesday March 30, 2021, 2:30PM, 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

Tuesday March 23, 2021, 2:30PM, Online

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

One world numeration seminar

Tuesday March 16, 2021, 2:30PM, 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

Tuesday March 9, 2021, 2:30PM, 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

Tuesday March 2, 2021, 4PM, Online

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

One world numeration seminar

Tuesday February 23, 2021, 2:30PM, Online

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

One world numeration seminar

Tuesday February 16, 2021, 2:30PM, Online

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

One world numeration seminar

Tuesday February 9, 2021, 2:30PM, Online

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

One world numeration seminar

Tuesday February 2, 2021, 2:30PM, 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

Tuesday January 26, 2021, 2:30PM, 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

Tuesday January 19, 2021, 2:30PM, 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

Tuesday January 5, 2021, 2:30PM, 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.

#### Year 2020

One world numeration seminar

Tuesday December 15, 2020, 2:30PM, 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

Tuesday December 8, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday December 1, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday November 17, 2020, 2:30PM, 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

Tuesday November 10, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday November 3, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday October 27, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday October 20, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday October 13, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday October 6, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday September 29, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday September 22, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday September 15, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday September 8, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday September 1, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday July 14, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday July 7, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday June 30, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday June 23, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday June 16, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday June 9, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday June 2, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday May 26, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday May 19, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday May 12, 2020, 2:30PM, Online

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

One world numeration seminar

Tuesday May 5, 2020, 2:30PM, Online

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