# One World Numeration Seminar

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

## Past talks

June 18, 2024
Noy Soffer Aranov (Technion) : Escape of Mass of the Thue Morse Sequence   (video) (slides)

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 θ, the degrees of the periodic part of the continued fraction of tn θ 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.

May 21, 2024
Gaétan Guillot (Université Paris-Saclay): Approximation of linear subspaces by rational linear subspaces   (video)

We elaborate on a problem raised by Schmidt in 1967: rational approximation of linear subspaces of Rn . 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.

May 7, 2024
Tom Kempton (University of Manchester): The Dynamics of the Fibonacci Partition Function   (video) (slides) (paper)

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).

April 23, 2024
Shunsuke Usuki (Kyoto University): On a lower bound of the number of integers in Littlewood's conjecture   (video) (slides) (paper1) (paper2)

Littlewood's conjecture is a famous and long-standing open problem which states that, for every (α,β) in R2, n ||nα|| ||nβ|| can be arbitrarily small for some integer n. This problem is closely related to the action of diagonal matrices on SL(3,R)/SL(3,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 (α,β) in R2 except on sets of small Hausdorff dimension, an estimate of the number of integers n which make n ||nα|| ||nβ|| small. The keys for the proof are the measure rigidity and further studies on behavior of empirical measures for the diagonal action.

March 26, 2024
Nikita Shulga (La Trobe University): Radical bound for Zaremba’s conjecture   (video) (slides) (paper)

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.

March 12, 2024
Joël Ouaknine (Max Planck Institute for Software Systems): The Skolem Landscape   (video) (slides)

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.

February 13, 2024
Bartosz Sobolewski (Jagiellonian University in Kraków and Montanuniversität Leoben): Block occurrences in the binary expansion of n and n+t   (video) (slides) (paper)

Let s(n) denote the sum of binary digits of a nonnegative integer n. In 2012 Cusick asked whether for every nonnegative integer t the set of n satisfying s(n+t) ≥ s(n) has natural density strictly greater than 1/2. So far it is known that the answer is affirmative for almost all t (in the sense of density) and s(n+t)-s(n) has approximately Gaussian distribution. During the talk we consider an analogue of this problem concerning the function r(n), which counts the occurrences of the block 11 in the binary expansion of n. In particular, we prove that the distribution of r(n+t)-r(n) is approximately Gaussian as well. We also discuss a generalization to an arbitrary block of binary digits. This is a joint work with Lukas Spiegelhofer.

January 30, 2024
Cathy Swaenepoel (Université Paris Cité): Reversible primes   (video) (slides) (paper)

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 rev(k) the reverse of k in base 2, defined by rev(k) = ∑0≤i<n εi 2n-1-i where k = ∑0≤i<n εi 2i with εi ∈ {0,1}, i ∈ {0,...,n−1}, εn−1 = 1. A natural question is to estimate the number of primes p ∈ [2n-1, 2n) such that rev(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 rev(k) have at most 8 prime factors (counted with multiplicity). We will also present an asymptotic formula for the number of integers k ∈ [2n-1, 2n) such that k and rev(k) are squarefree.
This is a joint work with Cécile Dartyge, Bruno Martin, Joël Rivat and Igor Shparlinski.

January 16, 2024
Karma Dajani (Universiteit Utrecht): Alternating N-continued fraction expansions   (video) (slides)

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 (N1, ..., Nm). 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.

December 12, 2023
Yasushi Nagai (Shinshu University): Overlap algorithm for general S-adic tilings   (video) (slides)

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.

November 28, 2023
Claudio Bonanno (Università di Pisa): Asymptotic behaviour of the sums of the digits for continued fraction algorithms   (video) (slides) (paper)

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.

November 14, 2023
Jana Lepšová (České vysoké učení technické v Praze, Université de Bordeaux): Dumont-Thomas numeration systems for ℤ   (slides) (paper)

We extend the well-known Dumont-Thomas numeration system to ℤ by considering two-sided periodic points of a substitution, thus allowing us to represent any integer in 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 of the periodic point when fed with the representation of n. The numeration system naturally extends to higher dimensions. 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 a Fibonacci analogue of the two's complement numeration system.

October 31, 2023
Stefano Marmi (Scuola Normale Superiore): Complexified continued fractions and complex Brjuno and Wilton functions   (video) (slides) (paper)

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: https://link.springer.com/article/10.1007/s00010-023-00967-w

October 17, 2023
Fumichika Takamizo (Osaka Metropolitan University): Finite β-expansion of natural numbers   (video) (slides) (paper)

If β is an integer, then each x ∈ Z[1/β] ∩ [0,∞) has finite expansion in base β. As a generalization of this property for β > 1, the condition (F1) that each x ∈ N has finite β-expansion was proposed by Frougny and Solomyak. In this talk, we give a sufficient condition for (F1). Moreover we also find β with property (F1) which does not have positive finiteness property.

October 3, 2023
Manfred Madritsch (Université de Lorraine): Construction of absolutely normal numbers   (video) (slides) (paper1) (paper2) (paper3)

Let b ≥ 2 be a positive integer. Then every real number x ∈ [0,1] admits a b-adic representation with digits ak. We call the real x simply normal to base b if every digit d ∈ {0,1,...,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, b2, b3, etc. Finally we call x absolutely normal if it is normal with respect to all bases b ≥ 2.
In the present talk we want to generalize this notion to normality in measure preserving systems like β-expansions and continued fraction expansions. Then we show constructions of numbers that are (absolutely) normal with respect to several different expansions.

September 19, 2023
James Worrell (University of Oxford): Transcendence of Sturmian Numbers over an Algebraic Base   (video) (slides) (paper)

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.

September 5, 2023
Mark Pollicott (University of Warwick): Complex Dimensions and Fractal Strings   (video) (slides)

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).

May 9, 2023
Craig S. Kaplan (University of Waterloo): An aperiodic monotile   (video) (slides) (paper)

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.

April 25, 2023
Ronnie Pavlov (University of Denver): Subshifts of very low complexity   (video) (slides) (paper)

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.

April 18, 2023
Anton Lukyanenko (George Mason University): Serendipitous decompositions of higher-dimensional continued fractions   (video) (pdf slides) (ppt slides) (paper1) (paper2) (paper3)

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 R3.
This is joint work with Joseph Vandehey.

March 28, 2023
Roland Zweimüller (Universität Wien): Variations on a theme of Doeblin   (video) (slides) (paper1) (paper2)

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.)

March 21, 2023
Demi Allen (University of Exeter): Diophantine Approximation for systems of linear forms - some comments on inhomogeneity, monotonicity, and primitivity   (video) (slides) (paper)

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 ψ-well-approximable number is one which can be approximated by rationals to a given degree of accuracy specified by an approximating function ψ. Khintchine's Theorem provides a beautiful characterisation of the Lebesgue measure of the set of ψ-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).

March 7, 2023
Derong Kong (Chongqing University): Critical values for the beta-transformation with a hole at 0   (video) (slides) (journal) (arXiv)

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.

February 14, 2023
Yining Hu (Huazhong University of Science and Technology): Algebraic automatic continued fractions in characteristic 2   (slides) (paper)

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

February 7, 2023
Ale Jan Homburg (Universiteit van Amsterdam, Vrije Universiteit Amsterdam): Iterated function systems of linear expanding and contracting maps on the unit interval   (video) (slides) (paper)

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.

January 31, 2023
Slade Sanderson (Universiteit Utrecht): Matching for parameterised symmetric golden maps   (video) (slides) (paper)

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 β, with β 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 β-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.

January 24, 2023
Kiko Kawamura (University of North Texas): The partial derivative of Okamoto's functions with respect to the parameter   (video) (slides) (paper)

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.

January 10, 2023
Roswitha Hofer (JKU Linz): Exact order of discrepancy of normal numbers   (video) (slides) (paper1) (paper2)

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 DN in N, a sequence ({bnα})n≥0, b ≥ 2 integer, can have for some real number α? If limN→∞ DN = 0 then α in called normal in base b.
So far the best upper bounds for DN for explicitly known normal numbers in base 2 are of the form NDN ≪ log2 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 NDN ≫ log2 N for Levin's binary normal number. So EITHER ND_N ≪ log2 N is the best possible order for DN in N of a normal number OR there exist another example of a binary normal number with a better growth of NDN in N. The recent result for Levin's normal number might support the conjecture that ND_N ≪ log2 N is the best order for DN in N a normal number can obtain.