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Tuesday at 2:30pm, online

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

Year 2024

One world numeration seminar
Tuesday June 18, 2024, 2PM, Online
Noy Soffer Aranov (Technion) Escape of Mass of the Thue Morse Sequence

One way to study the distribution of quadratic number fields is through the evolution of continued fraction expansions. In the function field setting, it was shown by de Mathan and Teullie that given a quadratic irrational $\Theta$, the degrees of the periodic part of the continued fraction of $t^n\Theta$ are unbounded. Paulin and Shapira improved this by proving that quadratic irrationals exhibit partial escape of mass. Moreover, they conjectured that they must exhibit full escape of mass. We show that the Thue Morse sequence is a counterexample to their conjecture. In this talk we shall discuss the technique of proof as well as the connection between escape of mass in continued fractions, Hecke trees, and number walls. This is part of ongoing work joint with Erez Nesharim.

One world numeration seminar
Tuesday May 21, 2024, 2PM, Online
Gaétan Guillot (Université Paris-Saclay) Approximation of linear subspaces by rational linear subspaces

We elaborate on a problem raised by Schmidt in 1967: rational approximation of linear subspaces of $\mathbb{R}^n$. In order to study the quality approximation of irrational numbers by rational ones, one can introduce the exponent of irrationality of a number. We can then generalize this notion in the framework of vector subspaces for the approximation of a subspace by so-called rational subspaces. After briefly introducing the tools for constructing this generalization, I will present the different possible studies of this object. Finally I will explain how we can construct spaces with prescribed exponents.

One world numeration seminar
Tuesday May 7, 2024, 2PM, Online
Tom Kempton (University of Manchester) The Dynamics of the Fibonacci Partition Function

The Fibonacci partition function $R(n)$ counts the number of ways of representing a natural number $n$ as the sum of distinct Fibonacci numbers. For example, $R(6)=2$ since $6=5+1$ and $6=3+2+1$. An explicit formula for $R(n)$ was recently given by Chow and Slattery. In this talk we express $R(n)$ in terms of ergodic sums over an irrational rotation, which allows us to prove lots of statements about the local structure of $R(n)$.

One world numeration seminar
Tuesday April 23, 2024, 2PM, Online
Shunsuke Usuki (Kyoto University) On a lower bound of the number of integers in Littlewood's conjecture

Littlewood's conjecture is a famous and long-standing open problem which states that, for every $(\alpha,\beta) \in \mathbb{R}^2$, $n\|n\alpha\|\|n\beta\|$ can be arbitrarily small for some integer $n$. This problem is closely related to the action of diagonal matrices on $\mathrm{SL}(3,\mathbb{R})/\mathrm{SL}(3,\mathbb{Z})$, and a groundbreaking result was shown by Einsiedler, Katok and Lindenstrauss from the measure rigidity for this action, saying that Littlewood's conjecture is true except on a set of Hausdorff dimension zero. In this talk, I will explain about a new quantitative result on Littlewood's conjecture which gives, for every $(\alpha,\beta) \in \mathbb{R}^2$ except on sets of small Hausdorff dimension, an estimate of the number of integers $n$ which make $n\|n\alpha\|\|n\beta\|$ small. The keys for the proof are the measure rigidity and further studies on behavior of empirical measures for the diagonal action.

One world numeration seminar
Tuesday April 9, 2024, 2PM, Online
Simon Kristensen (Aarhus Universitet) On the distribution of sequences of the form $(q_n y)$

The distribution of sequences of the form $(q_n y)$ with $(q_n)$ a sequence of integers and $y$ a real number have attracted quite a bit of attention, for instance due to their relation to inhomogeneous Littlewood type problems. In this talk, we will provide some results on the Lebesgue measure and Hausdorff dimension on the set of points in the unit interval approximated to a certain rate by points from such a sequence. A feature of our approach is that we obtain estimates even in the case when the sequence $(q_n)$ grows rather slowly. This is joint work with Tomas Persson.

One world numeration seminar
Tuesday March 26, 2024, 2PM, Online
Nikita Shulga (La Trobe University) Radical bound for Zaremba’s conjecture

Zaremba's conjecture states that for each positive integer $q$, there exists a coprime integer $a$, smaller than $q$, such that partial quotients in the continued fraction expansion of $a/q$ are bounded by some absolute constant. Despite major breakthroughs in the recent years, the conjecture is still open. In this talk I will discuss a new result towards Zaremba's conjecture, proving that for each denominator, one can find a numerator, such that partial quotients are bounded by the radical of the denominator, i.e. the product of distinct prime factors. This generalizes the result by Niederreiter and improves upon some results of Moshchevitin-Murphy-Shkredov.

One world numeration seminar
Tuesday March 12, 2024, 2PM, Online
Joël Ouaknine (Max Planck Institute for Software Systems) The Skolem Landscape

The Skolem Problem asks how to determine algorithmically whether a given linear recurrence sequence (such as the Fibonacci numbers) has a zero. It is a central question in dynamical systems and number theory, and has many connections to other branches of mathematics and computer science. Unfortunately, its decidability has been open for nearly a century! In this talk, I will present a survey of what is known on the Skolem Problem and related questions, including recent and ongoing developments.

One world numeration seminar
Tuesday January 30, 2024, 2PM, Online
Cathy Swaenepoel (IMJ-PRG) Reversible primes

The properties of the digits of prime numbers and various other sequences of integers have attracted great interest in recent years. For any positive integer $k$, we denote by $\overleftarrow{k}$ the reverse of $k$ in base 2, defined by $\overleftarrow{k} = \sum_{j=0}^{n-1} \varepsilon_j\,2^{n-1-j}$ where $k = \sum_{j=0}^{n-1} \varepsilon_{j} \,2^j$ with $\varepsilon_j \in \{0,1\}$, $j\in\{0, \ldots, n-1\}$, $ \varepsilon_{n-1} = 1$. A natural question is to estimate the number of primes $p\in \left[2^{n-1},2^n\right)$ such that $\overleftarrow{p}$ is prime. We will present a result which provides an upper bound of the expected order of magnitude. Our method is based on a sieve argument and also allows us to obtain a strong lower bound for the number of integers $k$ such that $k$ and $\overleftarrow{k}$ have at most 8 prime factors (counted with multiplicity). We will also present an asymptotic formula for the number of integers $k\in \left[2^{n-1},2^n\right)$ such that $k$ and $\overleftarrow{k}$ are squarefree.

This is a joint work with Cécile Dartyge, Bruno Martin, Joël Rivat and Igor Shparlinski.

One world numeration seminar
Tuesday January 16, 2024, 2PM, Online
Karma Dajani (Universiteit Utrecht) Alternating N-continued fraction expansions

We introduce a family of maps generating continued fractions where the digit 1 in the numerator is replaced cyclically by some given non-negative integers $(N_1, \dots, N_m)$. We prove the convergence of the given algorithm, and study the underlying dynamical system generating such expansions. We prove the existence of a unique absolutely continuous invariant ergodic measure. In special cases, we are able to build the natural extension and give an explicit expression of the invariant measure. For these cases, we formulate a Doeblin-Lenstra type theorem. For other cases we have a more implicit expression that we conjecture gives the invariant density. This conjecture is supported by simulations. For the simulations we use a method that gives us a smooth approximation in every iteration. This is joint work with Niels Langeveld.

Year 2023

One world numeration seminar
Tuesday December 12, 2023, 2PM, Online
Yasushi Nagai (Shinshu University) Overlap algorithm for general S-adic tilings

We investigate the question of when a tiling has pure point spectrum, for the class of S-adic tilings, which includes all self-affine tilings. The overlap algorithm by Solomyak is a powerful tool to study this problem for the class of self-affine tilings. We generalize this algorithm for general S-adic tilings, and apply it to a class of block S-adic tilings to show almost all of them have pure point spectra. This is a joint work with Jörg Thuswaldner.

One world numeration seminar
Tuesday November 28, 2023, 2PM, Online
Claudio Bonanno (Università di Pisa) Asymptotic behaviour of the sums of the digits for continued fraction algorithms

In this talk I will discuss applications of methods of ergodic theory to obtain pointwise asymptotic behaviour for the sum of the digits of some non-regular continued fraction algorithms. The idea is to study the behaviour of trimmed Birkhoff sums for infinite-measure preserving dynamical systems. The talk is based on joint work with Tanja I. Schindler.

One world numeration seminar
Tuesday November 14, 2023, 2PM, Online
Jana Lepšová (České vysoké učení technické v Praze, Université de Bordeaux) Dumont-Thomas numeration systems for ℤ

We extend the well-known Dumont-Thomas numeration system to $\mathbb{Z}$ by considering two-sided periodic points of a substitution, thus allowing us to represent any integer in $\mathbb{Z}$ by a finite word (starting with 0 when nonnegative and with 1 when negative). We show that an automaton returns the letter at position $n \in \mathbb{Z}$ of the periodic point when fed with the representation of $n$. The numeration system naturally extends to $\mathbb{Z}^d$. We give an equivalent characterization of the numeration system in terms of a total order on a regular language. Lastly, using particular periodic points, we recover the well-known two's complement numeration system and the Fibonacci analogue of the two's complement numeration system.

One world numeration seminar
Tuesday October 31, 2023, 2PM, Online
Stefano Marmi (Scuola Normale Superiore) Complexified continued fractions and complex Brjuno and Wilton functions

We study functions related to the classical Brjuno function, namely k-Brjuno functions and the Wilton function. Both appear in the study of boundary regularity properties of (quasi) modular forms and their integrals. We then complexify the functional equations which they fulfill and we construct analytic extensions of the k-Brjuno and Wilton functions to the upper half-plane. We study their boundary behaviour using an extension of the continued fraction algorithm to the complex plane. We also prove that the harmonic conjugate of the real k-Brjuno function is continuous at all irrational numbers and has a decreasing jump of π/qk at rational points p/q. This is based on joint work with S. B. Lee, I. Petrykiewicz and T. I. Schindler, the paper is available (open source) at this link:

One world numeration seminar
Tuesday October 17, 2023, 2PM, Online
Fumichika Takamizo (Osaka Metropolitan University) Finite $\beta$-expansion of natural numbers

If $\beta$ is an integer, then each $x \in \mathbb{Z}[1/\beta] \cap [0,\infty)$ has finite expansion in base $\beta$. As a generalization of this property for $\beta>1$, the condition (F$_{1}$) that each $x \in \mathbb{N}$ has finite $\beta$-expansion was proposed by Frougny and Solomyak. In this talk, we give a sufficient condition for (F$_{1}$). Moreover we also find $\beta$ with property (F$_{1}$) which does not have positive finiteness property.

One world numeration seminar
Tuesday October 3, 2023, 2PM, Online
Manfred Madritsch (Université de Lorraine) Construction of absolutely normal numbers

Let $b\geq2$ be a positive integer. Then every real number $x\in[0,1]$ admits a $b$-adic representation with digits $a_k$. We call the real $x$ simply normal to base $b$ if every digit $d\in\{0,1,\dots,b-1\}$ occurs with the same frequency in the $b$-ary representation. Furthermore we call $x$ normal to base $b$, if it is simply normal with respect to $b$, $b^2$, $b^3$, etc. Finally we call $x$ absolutely normal if it is normal with respect to all bases $b\geq2$.

In the present talk we want to generalize this notion to normality in measure preserving systems like $\beta$-expansions and continued fraction expansions. Then we show constructions of numbers that are (absolutely) normal with respect to several different expansions.

One world numeration seminar
Tuesday September 19, 2023, 2PM, Online
James Worrell (University of Oxford) Transcendence of Sturmian Numbers over an Algebraic Base

Ferenczi and Mauduit showed in 1997 that a number represented over an integer base by a Sturmian sequence of digits is transcendental. In this talk we generalise this result to hold for all algebraic number base b of absolute value strictly greater than one. More generally, for a given base b and given irrational number θ, we prove rational linear independence of the set comprising 1 together with all numbers of the above form whose associated digit sequences have slope θ.

We give an application of our main result to the theory of dynamical systems. We show that for a Cantor set C arising as the set of limit points of a contracted rotation f on the unit interval, where f is assumed to have an algebraic slope, all elements of C except its endpoints 0 and 1 are transcendental.

This is joint work with Florian Luca and Joel Ouaknine.

One world numeration seminar
Tuesday September 5, 2023, 2PM, Online
Mark Pollicott (University of Warwick) Complex Dimensions and Fractal Strings

Some years ago M.Lapidus introduced the notion of complex dimensions for a Cantor set in the real line. These occur as poles of the complex Dirichlet series formed from the lengths of the bounded intervals (the “fractal strings”) in the complement of the Cantor set. We will explore further these ideas when the Cantor set is the attractor of an iterated function scheme (concentrating on those whose contractions are a finite set of inverse branches of the usual Gauss map).

One world numeration seminar
Tuesday May 9, 2023, 2PM, Online
Craig S. Kaplan (University of Waterloo) An aperiodic monotile

A set of shapes is called aperiodic if the shapes admit tilings of the plane, but none that have translational symmetry. A longstanding open problem asks whether a set consisting of a single shape could be aperiodic; such a shape is known as an aperiodic monotile or sometimes an “einstein”. The recently discovered “hat” monotile settles this problem in two dimensions. In this talk I provide necessary background on aperiodicity and related topics in tiling theory, review the history of the search for for an aperiodic monotile, and then discuss the hat and its mathematical properties.

One world numeration seminar
Tuesday April 25, 2023, 3PM, Online
Ronnie Pavlov (University of Denver) Subshifts of very low complexity

The word complexity function $p(n)$ of a subshift $X$ measures the number of $n$-letter words appearing in sequences in $X$, and $X$ is said to have linear complexity if $p(n)/n$ is bounded. It's been known since work of Ferenczi that subshifts X with linear word complexity function (i.e. $\limsup p(n)/n$ finite) have highly constrained/structured behavior. I'll discuss recent work with Darren Creutz, where we show that if $\limsup p(n)/n < 4/3$, then the subshift $X$ must in fact have measurably discrete spectrum, i.e. it is isomorphic to a compact abelian group rotation. Our proof uses a substitutive/S-adic decomposition for such shifts, and I'll touch on connections to the so-called S-adic Pisot conjecture.

One world numeration seminar
Tuesday April 18, 2023, 2PM, Online
Anton Lukyanenko (George Mason University) Serendipitous decompositions of higher-dimensional continued fractions

Complex continued fractions (CFs) represent a complex number using a descending fraction with Gaussian integer coefficients. The associated dynamical system is exact (Nakada 1981) with a piecewise-analytic invariant measure (Hensley 2006). Certain higher-dimensional CFs, including CFs over quaternions, octonions, as well as the non-commutative Heisenberg group can be understood in a unified way using the Iwasawa CF framework (L-Vandehey 2022). Under some natural and robust assumptions, ergodicity of the associated systems can then be derived from a connection to hyperbolic geodesic flow, but stronger mixing results and information about the invariant measure remain elusive. Here, we study Iwasawa CFs under a more delicate serendipity assumption that yields the finite range condition, allowing us to extend the Nakada-Hensley results to certain Iwasawa CFs over the quaternions, octonions, and in $\mathbb{R}^3$.

This is joint work with Joseph Vandehey.

One world numeration seminar
Tuesday March 28, 2023, 2PM, Online
Roland Zweimüller (Universität Wien) Variations on a theme of Doeblin

Starting from Doeblin's observation on the Poissonian nature of occurrences of large digits in typical continued fraction expansions, I will outline some recent work on rare events in measure preserving systems (including spatiotemporal and local limit theorems) which, in particular, allows us to refine Doeblin's statement in several ways.

(Part of this is joint work with Max Auer.)

One world numeration seminar
Tuesday March 21, 2023, 2PM, Online
Demi Allen (University of Exeter) Diophantine Approximation for systems of linear forms - some comments on inhomogeneity, monotonicity, and primitivity

Diophantine Approximation is a branch of Number Theory in which the central theme is understanding how well real numbers can be approximated by rationals. In the most classical setting, a $\psi$-well-approximable number is one which can be approximated by rationals to a given degree of accuracy specified by an approximating function $\psi$. Khintchine's Theorem provides a beautiful characterisation of the Lebesgue measure of the set of $\psi$-well-approximable numbers and is one of the cornerstone results of Diophantine Approximation. In this talk I will discuss the generalisation of Khintchine's Theorem to the setting of approximation for systems of linear forms. I will focus mainly on the topic of inhomogeneous approximation for systems of linear forms. Time permitting, I may also discuss approximation for systems of linear forms subject to certain primitivity constraints. This talk will be based on joint work with Felipe Ramirez (Wesleyan, US).

One world numeration seminar
Tuesday March 7, 2023, 2PM, Online
Derong Kong (Chongqing University) Critical values for the beta-transformation with a hole at 0

Given $\beta \in (1,2]$, let $T$ be the $\beta$-transformation on the unit circle $[0,1)$. For $t \in [0,1)$ let $K(t)$ be the survivor set consisting of all $x$ whose orbit under $T$ never hits the open interval $(0,t)$. Kalle et al. [ETDS, 2020] proved that the Hausdorff dimension function $\dim K(t)$ is a non-increasing Devil's staircase in $t$. So there exists a critical value such that $\dim K(t)$ is vanishing when $t$ is passing through this critical value. In this paper we will describe this critical value and analyze its interesting properties. Our strategy to find the critical value depends on certain substitutions of Farey words and a renormalization scheme from dynamical systems. This is joint work with Pieter Allaart.

One world numeration seminar
Tuesday February 14, 2023, 2PM, Online
Yining Hu (Huazhong University of Science and Technology) Algebraic automatic continued fractions in characteristic 2

We present two families of automatic sequences that define algebraic continued fractions in characteristic $2$. The period-doubling sequence belongs to the first family $\mathcal{P}$; and its sum modulo $2$, the Thue-Morse sequence, belongs to the second family $\mathcal{G}$. The family $\mathcal{G}$ contains all the iterated sums of sequences from the $\mathcal{P}$ and more.

One world numeration seminar
Tuesday February 7, 2023, 2PM, Online
Ale Jan Homburg (Universiteit van Amsterdam, Vrije Universiteit Amsterdam) Iterated function systems of linear expanding and contracting maps on the unit interval

We analyze the two-point motions of iterated function systems on the unit interval generated by expanding and contracting affine maps, where the expansion and contraction rates are determined by a pair $(M,N)$ of integers.

This dynamics depends on the Lyapunov exponent.

For a negative Lyapunov exponent we establish synchronization, meaning convergence of orbits with different initial points. For a vanishing Lyapunov exponent we establish intermittency, where orbits are close for a set of iterates of full density, but are intermittently apart. For a positive Lyapunov exponent we show the existence of an absolutely continuous stationary measure for the two-point dynamics and discuss its consequences.

For nonnegative Lyapunov exponent and pairs $(M,N)$ that are multiplicatively dependent integers, we provide explicit expressions for absolutely continuous stationary measures of the two-point dynamics. These stationary measures are infinite $\sigma$-finite measures in the case of zero Lyapunov exponent.

This is joint work with Charlene Kalle.

One world numeration seminar
Tuesday January 31, 2023, 2PM, Online
Slade Sanderson (Universiteit Utrecht) Matching for parameterised symmetric golden maps

In 2020, Dajani and Kalle investigated invariant measures and frequencies of digits of signed binary expansions arising from a parameterised family of piecewise linear interval maps of constant slope 2. Central to their study was a property called ‘matching,’ where the orbits of the left and right limits of discontinuity points agree after some finite number of steps. We obtain analogous results for a parameterised family of ‘symmetric golden maps’ of constant slope $\beta$, with $\beta$ the golden mean. Matching is again central to our methods, though the dynamics of the symmetric golden maps are more delicate than the binary case. We characterize the matching phenomenon in our setting, present explicit invariant measures and frequencies of digits of signed $\beta$-expansions, and—time permitting—show further implications for a family of piecewise linear maps which arise as jump transformations of the symmetric golden maps.

Joint with Karma Dajani.

One world numeration seminar
Tuesday January 24, 2023, 2PM, Online
Kiko Kawamura (University of North Texas) The partial derivative of Okamoto's functions with respect to the parameter

Okamoto's functions were introduced in 2005 as a one-parameter family of self-affine functions, which are expressed by ternary expansion of x on the interval [0,1]. By changing the parameter, one can produce interesting examples: Perkins' nowhere differentiable function, Bourbaki-Katsuura function and Cantor's Devil's staircase function.

In this talk, we consider the partial derivative of Okomoto's functions with respect to the parameter a. We place a significant focus on a = 1/3 to describe the properties of a nowhere differentiable function K(x) for which the set of points of infinite derivative produces an example of a measure zero set with Hausdorff dimension 1.

This is a joint work with T. Mathis and M.Paizanis (undergraduate students) and N.Dalaklis (graduate student). The talk is very accessible and includes many computer graphics.

One world numeration seminar
Tuesday January 10, 2023, 2PM, Online
Roswitha Hofer (JKU Linz) Exact order of discrepancy of normal numbers

In the talk we discuss some previous results on the discrepancy of normal numbers and consider the still open question of Korobov: What is the best possible order of discrepancy $D_N$ in $N$, a sequence $(\{b^n\alpha\})_{n\geq 0}$, $b\geq 2,\in\mathbb{N}$, can have for some real number $\alpha$? If $\lim_{N\to\infty} D_N=0$ then $\alpha$ in called normal in base $b$.

So far the best upper bounds for $D_N$ for explicitly known normal numbers in base $2$ are of the form $ND_N\ll\log^2 N$. The first example is due to Levin (1999), which was later generalized by Becher and Carton (2019). In this talk we discuss the recent result in joint work with Gerhard Larcher that guarantees $ND_N\gg \log^2 N$ for Levin's binary normal number. So EITHER $ND_N\ll \log^2N$ is the best possible order for $D_N$ in $N$ of a normal number OR there exist another example of a binary normal number with a better growth of $ND_N$ in $N$. The recent result for Levin's normal number might support the conjecture that $ND_N\ll \log^2N$ is the best order for $D_N$ in $N$ a normal number can obtain.

Year 2022

One world numeration seminar
Tuesday December 13, 2022, 2PM, Online
Hiroki Takahasi (Keio University) Distribution of cycles for one-dimensional random dynamical systems

We consider an independently identically distributed random dynamical system generated by finitely many, non-uniformly expanding Markov interval maps with a finite number of branches. Assuming a topologically mixing condition and the uniqueness of equilibrium state for the associated skew product map, we establish a samplewise (quenched) almost-sure level-2 weighted equidistribution of “random cycles”, with respect to a natural stationary measure as the periods of the cycles tend to infinity. This result implies an analogue of Bowen's theorem on periodic orbits of topologically mixing Axiom A diffeomorphisms.

This talk is based on the preprint arXiv:2108.05522. If time permits, I will mention some future perspectives in this project.

One world numeration seminar
Tuesday December 6, 2022, 2PM, Online
Christoph Bandt (Universität Greifswald) Automata generated topological spaces and self-affine tilings

Numeration assigns symbolic sequences as addresses to points in a space X. There are points which get multiple addresses. It is known that these identifications describe the topology of X and can often be determined by an automaton. Here we define a corresponding class of automata and discuss their properties and interesting examples. Various open questions concern the realization of such automata by iterated functions and the uniqueness of such an implementation. Self-affine tiles form a simple class of examples.

One world numeration seminar
Tuesday November 29, 2022, 2PM, Online
Manuel Hauke (TU Graz) The asymptotic behaviour of Sudler products

Given an irrational number $\alpha$, we study the asymptotic behaviour of the Sudler product defined by $P_N(\alpha) = \prod_{r=1}^N 2 \lvert \sin \pi r \alpha \rvert$, which appears in many different areas of mathematics. In this talk, we explain the connection between the size of $P_N(\alpha)$ and the Ostrowski expansion of $N$ with respect to $\alpha$. We show that $\liminf_{N \to \infty} P_N(\alpha) = 0$ and $\limsup_{N \to \infty} P_N(\alpha)/N = \infty$, whenever the sequence of partial quotients in the continued fraction expansion of $\alpha$ exceeds $7$ infinitely often, and show that the value $7$ is optimal.

For Lebesgue-almost every $\alpha$, we can prove more: we show that for every non-decreasing function $\psi: (0,\infty) \to (0,\infty)$ with $\sum_{k=1}^{\infty} \frac{1}{\psi(k)} = \infty$ and $\liminf_{k \to \infty} \psi(k)/(k \log k)$ sufficiently large, the conditions $\log P_N(\alpha) \leq -\psi(\log N)$, $\log P_N(\alpha) \geq \psi(\log N)$ hold on sets of upper density $1$ respectively $1/2$.

One world numeration seminar
Tuesday November 22, 2022, 2PM, Online
Faustin Adiceam (Université Paris-Est Créteil) Badly approximable vectors and Littlewood-type problems

Badly approximable vectors are fractal sets enjoying rich Diophantine properties. In this respect, they play a crucial role in many problems well beyond Number Theory and Fractal Geometry (e.g., in signal processing, in mathematical physics and in convex geometry).

After outlining some of the latest developments in this very active area of research, we will take an interest in the Littlewood conjecture (c. 1930) and in its variants which all admit a natural formulation in terms of properties satisfied by badly approximable vectors. We will then show how ideas emerging from the mathematical theory of quasicrystals, from numeration systems and from the theory of aperiodic tilings have recently been used to refute the so-called t-adic Littlewood conjecture.

All necessary concepts will be defined in the talk. Joint with Fred Lunnon (Maynooth) and Erez Nesharim (Technion, Haifa).

One world numeration seminar
Tuesday November 15, 2022, 2PM, Online
Seul Bee Lee (Institute for Basic Science) Regularity properties of Brjuno functions associated with by-excess, odd and even continued fractions

An irrational number is called a Brjuno number if the sum of the series of $\log(q_{n+1})/q_n$ converges, where $q_n$ is the denominator of the $n$-th principal convergent of the regular continued fraction. The importance of Brjuno numbers comes from the study of one variable analytic small divisor problems. In 1988, J.-C. Yoccoz introduced the Brjuno function which characterizes the Brjuno numbers to estimate the size of Siegel disks. In this talk, we introduce Brjuno-type functions associated with by-excess, odd and even continued fractions with a number theoretical motivation. Then we discuss the $L^p$ and the Hölder regularity properties of the difference between the classical Brjuno function and the Brjuno-type functions. This is joint work with Stefano Marmi.

One world numeration seminar
Tuesday November 8, 2022, 2PM, Online
Wen Wu (South China University of Technology) From the Thue-Morse sequence to the apwenian sequences

In this talk, we will introduce a class of $\pm 1$ sequences, called the apwenian sequences. The Hankel determinants of these $\pm1$ sequences share the same property as the Hankel determinants of the Thue-Morse sequence found by Allouche, Peyrière, Wen and Wen in 1998. In particular, the Hankel determinants of apwenian sequences do not vanish. This allows us to discuss the Diophantine property of the values of their generating functions at $1/b$ where $b\geq 2$ is an integer. Moreover, the number of $\pm 1$ apwenian sequences is given explicitly. Similar questions are also discussed for $0$-$1$ apwenian sequences. This talk is based on joint work with Y.-J. Guo and G.-N. Han.

One world numeration seminar
Tuesday October 25, 2022, 2PM, Online
Álvaro Bustos-Gajardo (The Open University) Quasi-recognizability and continuous eigenvalues of torsion-free S-adic systems

We discuss combinatorial and dynamical descriptions of S-adic systems generated by sequences of constant-length morphisms between alphabets of bounded size. For this purpose, we introduce the notion of quasi-recognisability, a strictly weaker version of recognisability but which is indeed enough to reconstruct several classical arguments of the theory of constant-length substitutions in this more general context. Furthermore, we identify a large family of directive sequences, which we call “torsion-free”, for which quasi-recognisability is obtained naturally, and can be improved to actual recognisability with relative ease.

Using these notions we give S-adic analogues of the notions of column number and height for substitutions, including dynamical and combinatorial interpretations of each, and give a general characterisation of the maximal equicontinuous factor of the identified family of S-adic shifts, showing as a consequence that in this context all continuous eigenvalues must be rational. As well, we employ the tools developed for a first approach to the measurable case.

This is a joint work with Neil Mañibo and Reem Yassawi.

One world numeration seminar
Tuesday October 18, 2022, 2PM, Online
Yufei Chen (TU Delft) Matching of orbits of certain N-expansions with a finite set of digits

In this talk we consider a class of continued fraction expansions: the so-called $N$-expansions with a finite digit set, where $N\geq 2$ is an integer. For $N$ fixed they are steered by a parameter $\alpha\in (0,\sqrt{N}-1]$. For $N=2$ an explicit interval $[A,B]$ was determined, such that for all $\alpha\in [A,B]$ the entropy $h(T_{\alpha})$ of the underlying Gauss-map $T_{\alpha}$ is equal. In this paper we show that for all $N\in\mathbb{N}$, $N\geq 2$, such plateaux exist. In order to show that the entropy is constant on such plateaux, we obtain the underlying planar natural extension of the maps $T_{\alpha}$, the $T_{\alpha}$-invariant measure, ergodicity, and we show that for any two $\alpha,\alpha'$ from the same plateau, the natural extensions are metrically isomorphic, and the isomorphism is given explicitly. The plateaux are found by a property called matching.

One world numeration seminar
Tuesday October 11, 2022, 2PM, Online
Lukas Spiegelhofer (Montanuniversität Leoben) Primes as sums of Fibonacci numbers

We prove that the Zeckendorf sum-of-digits function of prime numbers, $z(p)$, is uniformly distributed in residue classes. The main ingredient that made this proof possible is the study of very sparse arithmetic subsequences of $z(n)$. In other words, we will meet the level of distribution. Our proof of this central result is based on a combination of the “Mauduit−Rivat−van der Corput method” for digital problems and an estimate of a Gowers norm related to $z(n)$. Our method of proof yields examples of substitutive sequences that are orthogonal to the Möbius function (cf. Sarnak's conjecture).

This is joint work with Michael Drmota and Clemens Müllner (TU Wien).

One world numeration seminar
Tuesday October 4, 2022, 2PM, Online
David Siukaev (Higher School of Economics) Exactness and Ergodicity of Certain Markovian Multidimensional Fraction Algorithms

A multidimensional continued fraction algorithm is a generalization of well-known continued fraction algorithms of small dimensions: Gauss and Euclidean. Ergodic properties of Markov MCF algorithms (ergodicity, nonsingularity, exactness, bi-measurability) affect their convergence (if the MСF algorithm is a Markov algorithm, there is a relationship between the spectral properties and its convergence).

In 2013 T. Miernowski and A. Nogueira proved that the Euclidean algorithm and the non-homogeneous Rauzy induction satisfy the intersection property and, as a consequence, are exact. At the end of the article it is stated that other non-homogeneous markovian algorithms (Selmer, Brun and Jacobi-Perron) also satisfy the intersection property and they also exact. However, there is no proof of this. In our paper this proof is obtained by using the structure of the proof of the exactness of the Euclidean algorithm with its generalization and refinement for multidimensional algorithms. We obtained technically complex proofs that differ from the proofs given in the article of T. Miernowski and A. Nogueira by the difficulties of generalization to the multidimensional case.

One world numeration seminar
Tuesday October 4, 2022, 2:30PM, Online
Alexandra Skripchenko (Higher School of Economics) Bruin-Troubetzkoy family of interval translation mappings: a new glance

In 2002 H. Bruin and S. Troubetzkoy described a special class of interval translation mappings on three intervals. They showed that in this class the typical ITM could be reduced to an interval exchange transformations. They also proved that generic ITM of their class that can not be reduced to IET is uniquely ergodic.

We suggest an alternative proof of the first statement and get a stronger version of the second one. It is a joint work in progress with Mauro Artigiani and Pascal Hubert.

One world numeration seminar
Tuesday September 27, 2022, 2PM, Online
Niels Langeveld (Montanuniversität Leoben) $N$-continued fractions and $S$-adic sequences

Given the $N$-continued fraction of a number $x$, we construct $N$-continued fraction sequences in the same spirit as Sturmian sequences can be constructed from regular continued fractions. These sequences are infinite words over a two letter alphabet obtained as the limit of a directive sequence of certain substitutions (they are S-adic sequences). By viewing them as a generalisation of Sturmian sequences it is natural to study balancedness. We will see that the sequences we construct are not 1-balanced but C-balanced for $C=N^2$. Furthermore, we construct a dual sequence which is related to the natural extension of the $N$-continued fraction algorithm. This talk is joint work with Lucía Rossi and Jörg Thuswaldner.

One world numeration seminar
Tuesday September 13, 2022, 2:30PM, Online
Benedict Sewell (Alfréd Rényi Institute) An upper bound on the box-counting dimension of the Rauzy gasket

The Rauzy gasket is a subset of the standard two-simplex, and an important subset of parameter space in various settings. It is a parabolic, non-conformal fractal attractor; meaning that even the most trivial upper bounds on its Hausdorff or box-counting dimensions are hard to obtain. In this talk (featuring joint work with Mark Pollicott), we discuss how an elementary method leads to the best known upper bound on these dimensions.

One world numeration seminar
Tuesday July 12, 2022, 2:30PM, Online
Ruofan Li (South China University of Technology) Rational numbers in ×b-invariant sets

Let $b \ge 2$ be an integer and $S$ be a finite non-empty set of primes not containing divisors of $b$. For any $\times b$-invariant, non-dense subset $A$ of $[0,1)$, we prove the finiteness of rational numbers in $A$ whose denominators can only be divided by primes in $S$. A quantitative result on the largest prime divisors of the denominators of rational numbers in $A$ is also obtained. This is joint work with Bing Li and Yufeng Wu.

One world numeration seminar
Tuesday July 5, 2022, 2:30PM, Online
Charlene Kalle (Universiteit Leiden) Random Lüroth expansions

Since the introduction of Lüroth expansions by Lüroth in his paper from 1883 many results have appeared on their approximation properties. In 1990 Kalpazidou, Knopfmacher and Knopfmacher introduced alternating Lüroth expansions and studied their properties. A comparison between the two and other comparable number systems was then given by Barrionuevo, Burton, Dajani and Kraaikamp in 1996. In this talk we introduce a family of random dynamical systems that produce many Lüroth type expansions at once. Topics that we consider are periodic expansions, universal expansions, speed of convergence and approximation coefficients. This talk is based on joint work with Marta Maggioni.

One world numeration seminar
Tuesday June 21, 2022, 2:30PM, Online
James A. Yorke (University of Maryland) Large and Small Chaos Models

To set the scene, I will discuss one large model, a whole-Earth model for predicting the weather, and how to initialize such a model and what aspects of chaos are essential. Then I will discuss a couple related “very simple” maps that tell us a great deal about very complex models. The results on simple models are new. I will discuss the logistic map mx(1-x). Its dynamics can make us rethink climate models. Also, we have created a piecewise linear map on a 3D cube that is unstable in 2 dimensions in some places and unstable in 1 in others. It has a dense set of periodic points that are 1 D unstable and another dense set of periodic points that are all 2 D unstable. I will also discuss a new project whose tentative title is “ Can the flap of butterfly's wings shift a tornado out of Texas – without chaos?

One world numeration seminar
Tuesday June 7, 2022, 2:30PM, Online
Sophie Morier-Genoud (Université Reims Champagne Ardenne) q-analogues of real numbers

Classical sequences of numbers often lead to interesting q-analogues. The most popular among them are certainly the q-integers and the q-binomial coefficients which both appear in various areas of mathematics and physics. With Valentin Ovsienko we recently suggested a notion of q-rationals based on combinatorial properties and continued fraction expansions. The definition of q-rationals naturally extends the one of q-integers and leads to a ratio of polynomials with positive integer coefficients. I will explain the construction and give the main properties. In particular I will briefly mention connections with the combinatorics of posets, cluster algebras, Jones polynomials, homological algebra. Finally I will also present further developments of the theory, leading to the notion of q-irrationals and q-unimodular matrices.

One world numeration seminar
Tuesday May 31, 2022, 2:30PM, Online
Verónica Becher (Universidad de Buenos Aires & CONICET Argentina) Poisson generic real numbers

Years ago Zeev Rudnick defined the Poisson generic real numbers as those where the number of occurrences of the long strings in the initial segments of their fractional expansions in some base have the Poisson distribution. Yuval Peres and Benjamin Weiss proved that almost all real numbers, with respect to Lebesgue measure, are Poisson generic. They also showed that Poisson genericity implies Borel normality but the two notions do not coincide, witnessed by the famous Champernowne constant. We recently showed that there are computable Poisson generic real numbers and that all Martin-Löf real numbers are Poisson generic.

This is joint work Nicolás Álvarez and Martín Mereb.

One world numeration seminar
Tuesday May 24, 2022, 2:30PM, Online
Émilie Charlier (Université de Liège) Spectrum, algebraicity and normalization in alternate bases

The first aim of this work is to give information about the algebraic properties of alternate bases determining sofic systems. We exhibit two conditions: one necessary and one sufficient. Comparing the setting of alternate bases to that of one real base, these conditions exhibit a new phenomenon: the bases should be expressible as rational functions of their product. The second aim is to provide an analogue of Frougny's result concerning normalization of real bases representations. Under some suitable condition (i.e., our previous sufficient condition for being a sofic system), we prove that the normalization function is computable by a finite Büchi automaton, and furthermore, we effectively construct such an automaton. An important tool in our study is the spectrum of numeration systems associated with alternate bases. For our purposes, we use a generalized concept of spectrum associated with a complex base and complex digits, and we study its topological properties.

This is joint work with Célia Cisternino, Zuzana Masáková and Edita Pelantová.

One world numeration seminar
Tuesday May 17, 2022, 2:30PM, Online
Vilmos Komornik (Université de Strasbourg et Shenzhen University) Topology of univoque sets in real base expansions

We report on a recent joint paper with Martijn de Vries and Paola Loreti. Given a positive integer $M$ and a real number $1 < q\le M+1$, an expansion of a real number $x \in \left[0,M/(q-1)\right]$ over the alphabet $A=\{0,1,\ldots,M\}$ is a sequence $(c_i) \in A^{\mathbb{N}}$ such that $x=\sum_{i=1}^{\infty}c_iq^{-i}$. Generalizing many earlier results, we investigate the topological properties of the set $U_q$ consisting of numbers $x$ having a unique expansion of this form, and the combinatorial properties of the set $U_q'$ consisting of their corresponding expansions.

One world numeration seminar
Tuesday May 3, 2022, 2:30PM, Online
Nicolas Chevallier (Université de Haute Alsace) Best Diophantine approximations in the complex plane with Gaussian integers

Starting with the minimal vectors in lattices over Gaussian integers in $\mathbb{C}^2$, we define a algorithm that finds the sequence of minimal vectors of any unimodular lattice in $\mathbb{C}^2$. Restricted to lattices associated with complex numbers this algorithm find all the best Diophantine approximations of a complex numbers. Following Doeblin, Lenstra, Bosma, Jager and Wiedijk, we study the limit distribution of the sequence of products $(u_{n1}u_{n2})_n$ where $(u_n=( u_{n1},u_{n2} ))_n$ is the sequence of minimal vectors of a lattice in $\mathbb{C}^2$. We show that there exists a measure in $\mathbb{C}$ which is the limit distribution of the sequence of products of almost all unimodular lattices.

One world numeration seminar
Tuesday April 19, 2022, 2:30PM, Online
Paulina Cecchi Bernales (Universidad de Chile) Coboundaries and eigenvalues of finitary S-adic systems

An S-adic system is a shift space obtained by performing an infinite composition of morphisms defined over possibly different finite alphabets. It is said to be finitary if these morphisms are taken from a finite set. S-adic systems are a generalization of substitution shifts. In this talk we will discuss spectral properties of finitary S-adic systems. Our departure point will be a theorem by B. Host which characterizes eigenvalues of substitution shifts, and where coboundaries appear as a key tool. We will introduce the notion of S-adic coboundaries and present some results which show how they are related with eigenvalues of S-adic systems. We will also present some applications of our results to constant-length finitary S-adic systems.

This is joint work with Valérie Berthé and Reem Yassawi.

One world numeration seminar
Tuesday April 12, 2022, 2:30PM, Online
Eda Cesaratto (Univ. Nac. de Gral. Sarmiento & CONICET, Argentina) Lochs-type theorems beyond positive entropy

Lochs' theorem and its generalizations are conversion theorems that relate the number of digits determined in one expansion of a real number as a function of the number of digits given in some other expansion. In its original version, Lochs' theorem related decimal expansions with continued fraction expansions. Such conversion results can also be stated for sequences of interval partitions under suitable assumptions, with results holding almost everywhere, or in measure, involving the entropy. This is the viewpoint we develop here. In order to deal with sequences of partitions beyond positive entropy, this paper introduces the notion of log-balanced sequences of partitions, together with their weight functions. These are sequences of interval partitions such that the logarithms of the measures of their intervals at each depth are roughly the same. We then state Lochs-type theorems which work even in the case of zero entropy, in particular for several important log-balanced sequences of partitions of a number-theoretic nature.

This is joint work with Valérie Berthé (IRIF), Pablo Rotondo (U. Gustave Eiffel) and Martín Safe (Univ. Nac. del Sur & CONICET, Argentina).

One world numeration seminar
Tuesday April 5, 2022, 2:30PM, Online
Jungwon Lee (University of Warwick) Dynamics of Ostrowski skew-product: Limit laws and Hausdorff dimensions

We discuss a dynamical study of the Ostrowski skew-product map in the context of inhomogeneous Diophantine approximation. We plan to outline the setup/ strategy based on transfer operator analysis and applications in arithmetic of number fields (joint with Valérie Berthé).

One world numeration seminar
Tuesday March 29, 2022, 2:30PM, Online
Tingyu Zhang (East China Normal University) Random β-transformation on fat Sierpiński gasket

We define the notions of greedy, lazy and random transformations on fat Sierpiński gasket. We determine the bases, for which the system has a unique measure of maximal entropy and an invariant measure of product type, with one coordinate being absolutely continuous with respect to Lebesgue measure.

This is joint work with K. Dajani and W. Li.

One world numeration seminar
Tuesday March 15, 2022, 2:30PM, Online
Pierre Popoli (Université de Lorraine) Maximum order complexity for some automatic and morphic sequences along polynomial values

Automatic sequences are not suitable sequences for cryptographic applications since both their subword complexity and their expansion complexity are small, and their correlation measure of order 2 is large. These sequences are highly predictable despite having a large maximum order complexity. However, recent results show that polynomial subsequences of automatic sequences, such as the Thue-Morse sequence or the Rudin-Shapiro sequence, are better candidates for pseudorandom sequences. A natural generalization of automatic sequences are morphic sequences, given by a fixed point of a prolongeable morphism that is not necessarily uniform. In this talk, I will present my results on lowers bounds for the maximum order complexity of the Thue-Morse sequence, the Rudin-Shapiro sequence and the sum of digits function in Zeckendorf base, which are respectively automatics and morphic sequences.

One world numeration seminar
Tuesday March 8, 2022, 2:30PM, Online
Michael Coons (Universität Bielefeld) A spectral theory of regular sequences

A few years ago, Michael Baake and I introduced a probability measure associated to Stern’s diatomic sequence, an example of a regular sequence—sequences which generalise constant length substitutions to infinite alphabets. In this talk, I will discuss extensions of these results to more general regular sequences as well as further properties of these measures. This is joint work with several people, including Michael Baake, James Evans, Zachary Groth and Neil Manibo.

One world numeration seminar
Tuesday March 1, 2022, 2:30PM, Online
Daniel Krenn (Universität Salzburg) k-regular sequences: Asymptotics and Decidability

A sequence $x(n)$ is called $k$-regular, if the set of subsequences $x(k^j n + r)$ is contained in a finitely generated module. In this talk, we will consider the asymptotic growth of $k$-regular sequences. When is it possible to compute it? …and when not? If possible, how precisely can we compute it? If not, is it just a lack of methods or are the underlying decision questions recursively solvable (i.e., decidable in a computational sense)? We will discuss answers to these questions. To round off the picture, we will consider further decidability questions around $k$-regular sequences and the subclass of $k$-automatic sequences.

One world numeration seminar
Tuesday February 15, 2022, 2:30PM, Online
Wolfgang Steiner (IRIF) Unique double base expansions

For pairs of real bases $\beta_0, \beta_1 > 1$, we study expansions of the form $\sum_{k=1}^\infty i_k / (\beta_{i_1} \beta_{i_2} \cdots \beta_{i_k})$ with digits $i_k \in \{0,1\}$. We characterise the pairs admitting non-trivial unique expansions as well as those admitting uncountably many unique expansions, extending recent results of Neunh\“auserer (2021) and Zou, Komornik and Lu (2021). Similarly to the study of unique $\beta$-expansions with three digits by the speaker (2020), this boils down to determining the cardinality of binary shifts defined by lexicographic inequalities. Labarca and Moreira (2006) characterised when such a shift is empty, at most countable or uncountable, depending on the position of the lower and upper bounds with respect to Thue–Morse–Sturmian words.
This is joint work with Vilmos Komornik and Yuru Zou.

One world numeration seminar
Tuesday February 8, 2022, 2:30PM, Online
Magdaléna Tinková (České vysoké učení technické v Praze) Universal quadratic forms, small norms and traces in families of number fields

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.

One world numeration seminar
Tuesday February 1, 2022, 2:30PM, Online
Jonas Jankauskas (Vilniaus universitetas) Digit systems with rational base matrix over lattices

Let $A$ be a matrix with rational entries and no eigenvalue in absolute value smaller than 1. Let $\mathbb{Z}^d[A]$ be the minimal $A$-invariant $\mathbb{Z}$-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 $\mathbb{Z}^d[A]$ has a finite radix expansion in base $A$ using only the digits from $D$, i.e. $\mathbb{Z}^d[A]=D[A]$. 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 $\mathbb{Z}^d$ and can be extended to $\mathbb{Z}^d[A]$. This is joint work with J. Thuswaldner, “Rational Matrix Digit Systems”, to appear in “Linear and Multilinear Algebra” (arXiv preprint:

One world numeration seminar
Tuesday January 25, 2022, 2:30PM, Online
Claudio Bonanno (Università di Pisa) Infinite ergodic theory and a tree of rational pairs

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.

One world numeration seminar
Tuesday January 18, 2022, 2:30PM, Online
Agamemnon Zafeiropoulos (NTNU) The order of magnitude of Sudler products

Given an irrational $\alpha \in [0,1] \smallsetminus \mathbb{Q}$, we define the corresponding Sudler product by $$ P_N(\alpha) = \prod_{n=1}^{N}2|\sin (\pi n \alpha)|. $$ In joint work with C. Aistleitner and N. Technau, we show that when $\alpha = [0;b,b,b…]$ is a quadratic irrational with all partial quotients in its continued fraction expansion equal to some integer b, the following hold:
- If $b\leq 5$, 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
Tuesday January 11, 2022, 2:30PM, Online
Philipp Gohlke (Universität Bielefeld) Zero measure spectrum for multi-frequency Schrödinger operators

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).

Year 2021

One world numeration seminar
Tuesday December 21, 2021, 2:30PM, Online
Fan Lü (Sichuan Normal University) Multiplicative Diophantine approximation in the parameter space of beta-dynamical system

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 $T_{\beta}$ be the beta-transformation with $\beta>1$ and $x$ be a fixed point in $(0,1]$, we consider the set of parameters $(\alpha, \beta)$, such that the multiple $\|T^n_{\alpha}(x)\|\|T^n_{\beta}(x)\|$ 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.

One world numeration seminar
Tuesday December 7, 2021, 2:30PM, Online
Jamie Walton (University of Nottingham) Extending the theory of symbolic substitutions to compact alphabets

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.

One world numeration seminar
Tuesday November 23, 2021, 2:30PM, Online
Sascha Troscheit (Universität Wien) Analogues of Khintchine's theorem for random attractors

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.

One world numeration seminar
Tuesday November 16, 2021, 2:30PM, Online
Lucía Rossi (Montanuniversität Leoben) Rational self-affine tiles associated to (nonstandard) digit systems

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.

One world numeration seminar
Tuesday November 9, 2021, 2:30PM, Online
Zhiqiang Wang (East China Normal University) How inhomogeneous Cantor sets can pass a point

Abstract: For $x > 0$, we define $\Upsilon(x) = \{ (a,b): x\in E_{a,b}, a>0, b>0, a+b \le 1 \}$, where the set $E_{a,b}$ is the unique nonempty compact invariant set generated by the inhomogeneous IFS $\{ f_0(x) = a x, f_1(x) = b(x+1) \}$. We show the set $\Upsilon(x)$ is a Lebesgue null set with full Hausdorff dimension in $\mathbb{R}^2$, and the intersection of sets $\Upsilon(x_1), \Upsilon(x_2), \cdots, \Upsilon(x_\ell)$ still has full Hausdorff dimension $\mathbb{R}^2$ for any finitely many positive real numbers $x_1, x_2, \cdots, x_\ell$.

One world numeration seminar
Tuesday November 9, 2021, 3PM, Online
Younès Tierce (Université de Rouen Normandie) Extensions of the random beta-transformation

Let $\beta \in (1,2)$ and $I_\beta := [0,\frac{1}{\beta-1}]$. Almost every real number of $I_\beta$ has infinitely many expansions in base $\beta$, and the random $\beta$-transformation generates all these expansions. We present the construction of a “geometrico-symbolic” extension of the random $\beta$-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.

One world numeration seminar
Tuesday November 2, 2021, 2:30PM, Online
Pieter Allaart (University of North Texas) On the existence of Trott numbers relative to multiple bases

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.

One world numeration seminar
Tuesday October 26, 2021, 2:30PM, Online
Michael Baake (Universität Bielefeld) Spectral aspects of aperiodic dynamical systems

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.

One world numeration seminar
Tuesday October 19, 2021, 2:30PM, Online
Mélodie Lapointe (IRIF) q-analog of the Markoff injectivity conjecture

The Markoff injectivity conjecture states that $w\mapsto\mu(w)_{12}$ is injective on the set of Christoffel words where $\mu:\{\mathtt{0},\mathtt{1}\}^*\to\mathrm{SL}_2(\mathbb{Z})$ is a certain homomorphism and $M_{12}$ is the entry above the diagonal of a $2\times2$ matrix $M$. Recently, Leclere and Morier-Genoud (2021) proposed a $q$-analog $\mu_q$ of $\mu$ such that $\mu_{q\to1}(w)_{12}=\mu(w)_{12}$ is the Markoff number associated to the Christoffel word $w$. We show that there exists an order $<_{radix}$ on $\{\mathtt{0},\mathtt{1}\}^*$ such that for every balanced sequence $s \in \{\mathtt{0},\mathtt{1}\}^\mathbb{Z}$ and for all factors $u, v$ in the language of $s$ with $u <_{radix} v$, the difference $\mu_q(v)_{12} - \mu_q(u)_{12}$ is a nonzero polynomial of indeterminate $q$ with nonnegative integer coefficients. Therefore, for every $q>0$, the map $\{\mathtt{0},\mathtt{1}\}^*\to\mathbb{R}$ defined by $w\mapsto\mu_q(w)_{12}$ 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.

One world numeration seminar
Tuesday October 12, 2021, 2:30PM, Online
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.

One world numeration seminar
Tuesday October 5, 2021, 2:30PM, Online
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).

One world numeration seminar
Tuesday October 5, 2021, 3PM, Online
Taylor Jones (University of North Texas) On the Existence of Numbers with Matching Continued Fraction and Decimal Expansion

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.

One world numeration seminar
Tuesday September 28, 2021, 2:30PM, Online
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.

One world numeration seminar
Tuesday September 21, 2021, 2:30PM, Online
Maria Siskaki (University of Illinois at Urbana-Champaign) The distribution of reduced quadratic irrationals arising from continued fraction expansions

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.

One world numeration seminar
Tuesday September 14, 2021, 2:30PM, Online
Steve Jackson (University of North Texas) Descriptive complexity in numeration systems

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, ????-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.

One world numeration seminar
Tuesday September 7, 2021, 2:30PM, Online
Oleg Karpenkov (University of Liverpool) On Hermite's problem, Jacobi-Perron type algorithms, and Dirichlet groups

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.

One world numeration seminar
Tuesday July 6, 2021, 2:30PM, Online
Niclas Technau (University of Wisconsin - Madison) Littlewood and Duffin-Schaeffer-type problems in diophantine approximation

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.

One world numeration seminar
Tuesday June 29, 2021, 2:30PM, Online
Polina Vytnova (University of Warwick) Hausdorff dimension of Gauss-Cantor sets and their applications to the study of classical Markov spectrum

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.

One world numeration seminar
Tuesday June 22, 2021, 2:30PM, Online
Lingmin Liao (Université Paris-Est Créteil Val de Marne) Simultaneous Diophantine approximation of the orbits of the dynamical systems x2 and x3

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.

One world numeration seminar
Tuesday June 15, 2021, 2:30PM, Online
Sam Chow (University of Warwick) Dyadic approximation in the Cantor set

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.

One world numeration seminar
Tuesday June 8, 2021, 2:30PM, Online
Shigeki Akiyama (University of Tsukuba) Counting balanced words and related problems

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.

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

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.

One world numeration seminar
Tuesday May 25, 2021, 2:30PM, Online
Charles Fougeron (IRIF) Dynamics of simplicial systems and multidimensional continued fraction algorithms

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.

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

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.

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 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.

One world numeration seminar
Tuesday May 4, 2021, 4PM, Online
Tushar Das (University of Wisconsin - La Crosse) Hausdorff Hensley Good & Gauss

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 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.

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

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.

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

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.

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

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.

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

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.

One world numeration seminar
Tuesday March 16, 2021, 2:30PM, Online
Alexandra Skripchenko (Higher School of Economics) Double rotations and their ergodic properties

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.

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

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.

One world numeration seminar
Tuesday March 2, 2021, 4PM, Online
Vitaly Bergelson (Ohio State University) Normal sets in (ℕ,+) and (ℕ,×)

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.

One world numeration seminar
Tuesday February 23, 2021, 2:30PM, Online
Seulbee Lee (Scuola Normale Superiore di Pisa) Odd-odd continued fraction algorithm

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.

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

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.

One world numeration seminar
Tuesday February 9, 2021, 2:30PM, Online
Clemens Müllner (TU Wien) Multiplicative automatic sequences

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.

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

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.

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

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.

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

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

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.

Year 2020

One world numeration seminar
Tuesday December 15, 2020, 2:30PM, Online
Lukas Spiegelhofer (Montanuniversität Leoben) The digits of n+t

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).

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

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 bn 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.

One world numeration seminar
Tuesday December 1, 2020, 2:30PM, Online
Michael Barnsley (Australian National University) Rigid fractal tilings

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.

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

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).

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

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).

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

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 (beta, 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.

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

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.

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

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.

One world numeration seminar
Tuesday October 13, 2020, 2:30PM, Online
Kan Jiang (Ningbo University) Representations of real numbers on fractal sets

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.

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

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.

One world numeration seminar
Tuesday September 29, 2020, 2:30PM, Online
Marta Maggioni (Leiden University) Random matching for random interval maps

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.

One world numeration seminar
Tuesday September 22, 2020, 2:30PM, Online
Yotam Smilansky (Rutgers University) Multiscale Substitution Tilings

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.

One world numeration seminar
Tuesday September 15, 2020, 2:30PM, Online
Jeffrey Shallit (University of Waterloo) Lazy Ostrowski Numeration and Sturmian Words

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.

One world numeration seminar
Tuesday September 8, 2020, 2:30PM, Online
Bing Li (South China University of Technology) Some fractal problems in beta-expansions

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.

One world numeration seminar
Tuesday September 1, 2020, 2:30PM, Online
Bill Mance (Adam Mickiewicz University in Poznań) Hotspot Lemmas for Noncompact Spaces

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.

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

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.

One world numeration seminar
Tuesday July 7, 2020, 2:30PM, Online
Hajime Kaneko (University of Tsukuba) Analogy of Lagrange spectrum related to geometric progressions

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.

One world numeration seminar
Tuesday June 30, 2020, 2:30PM, Online
Niels Langeveld (Leiden University) Continued fractions with two non integer digits

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 α1 and α2 with α1α2 ≤ 1/2 . If α1α2 < 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.

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

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).

One world numeration seminar
Tuesday June 16, 2020, 2:30PM, Online
Carlos Matheus (CNRS, École Polytechnique) Approximations of the Lagrange and Markov spectra

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.

One world numeration seminar
Tuesday June 9, 2020, 2:30PM, Online
Simon Baker (University of Birmingham) Equidistribution results for self-similar measures

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.

One world numeration seminar
Tuesday June 2, 2020, 2:30PM, Online
Henna Koivusalo (University of Vienna) Linear repetition in polytopal cut and project sets

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.

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

We consider a p-tuple of real numbers greater than 1, beta=(beta_1,…,beta_p), called an alternate base, to represent real numbers. Since these representations generalize the beta-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 beta-expansion and the (beta_1*…*beta_p)-expansion over a particular digit set and study the cases when the equality holds. Next, we will talk about the existence of a measure equivalent to Lebesgue, invariant for the transformation corresponding to the alternate base and also about the ergodicity of this transformation. This is a joint work with Émilie Charlier and Karma Dajani.

One world numeration seminar
Tuesday May 19, 2020, 2:30PM, Online
Boris Solomyak (University of Bar-Ilan) On singular substitution Z-actions

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.

One world numeration seminar
Tuesday May 12, 2020, 2:30PM, Online
Olivier Carton (Université de Paris) Preservation of normality by selection

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.

One world numeration seminar
Tuesday May 5, 2020, 2:30PM, Online
Narad Rampersad (University of Winnipeg) Ostrowski numeration and repetitions in words

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.