# One World Numeration Seminar

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

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September 27, 2022, 14:00 CEST (UTC +2)

Niels Langeveld (Montanuniversität Leoben): N-continued fractions and S-adic sequences (slides) (paper)

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.

October 4, 2022, 14:00 CEST (UTC +2)

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.

October 4, 2022, 14:30 CEST (UTC +2)

Alexandra Skripchenko (Higher School of Economics): TBA

October 11, 2022, 14:00 CEST (UTC +2)

Lukas Spiegelhofer (Montanuniversität Leoben): TBA

October 25, 2022, 14:00 CEST (UTC +2)

Álvaro Bustos-Gajardo (The Open University): TBA

## Past talks

September 13, 2022

Benedict Sewell (Alfréd Rényi Institute): An upper bound on the box-counting dimension of the Rauzy gasket (video) (slides) (paper)

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.

July 12, 2022

Ruofan Li (South China University of Technology): Rational numbers in ×b-invariant sets (video) (slides) (paper)

Let b ≥ 2 be an integer and S be a finite non-empty set of primes not containing divisors of b. For any ×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.

July 5, 2022

Charlene Kalle (Universiteit Leiden): Random Lüroth expansions (video) (slides) (journal) (arXiv)

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.

June 21, 2022

James A. Yorke (University of Maryland): Large and Small Chaos Models (video) (slides)

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?

June 7, 2022

Sophie Morier-Genoud (Université Reims Champagne Ardenne): q-analogues of real numbers (video) (paper1) (paper2) (paper3) (paper4)

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.

May 31, 2022

Verónica Becher (Universidad de Buenos Aires & CONICET Argentina): Poisson generic real numbers (slides) (paper)

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.

May 24, 2022

Émilie Charlier (Université de Liège): Spectrum, algebraicity and normalization in alternate bases (video) (slides) (paper)

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.

May 17, 2022

Vilmos Komornik (Shenzhen University and Université de Strasbourg): Topology of univoque sets in real base expansions (video) (slides) (paper)

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 ≤ M+1, an expansion of a real number x ∈ [0,M/(q-1)] over the alphabet A = {0,1,...,M} is a sequence (c_{i}) ∈ A^{N} such that x = Σ_{k=1}^{∞} c_{i} q^{-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.

May 3, 2022

Nicolas Chevallier (Université de Haute Alsace): Best Diophantine approximations in the complex plane with Gaussian integers (video) (slides) (journal) (arXiv)

Starting with the minimal vectors in lattices over Gaussian integers in C^{2}, we define a algorithm that finds the sequence of minimal vectors of any unimodular lattice in 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 C^{2}. We show that there exists a measure in C which is the limit distribution of the sequence of products of almost all unimodular lattices.

April 19, 2022

Paulina Cecchi Bernales (Universidad de Chile): Coboundaries and eigenvalues of finitary S-adic systems (video) (slides) (paper)

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.

April 12, 2022

Eda Cesaratto (Univ. Nac. de Gral. Sarmiento & CONICET, Argentina): Lochs-type theorems beyond positive entropy (video) (slides) (paper)

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).

April 5, 2022

Jungwon Lee (University of Warwick): Dynamics of Ostrowski skew-product: Limit laws and Hausdorff dimensions (video) (slides) (paper)

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).

March 29, 2022

Tingyu Zhang (East China Normal University): Random β-transformation on fat Sierpiński gasket (video) (slides) (paper)

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.

March 15, 2022

Pierre Popoli (Université de Lorraine): Maximum order complexity for some automatic and morphic sequences along polynomial values (video) (slides) (paper1) (paper2)

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.

March 8, 2022

Michael Coons (Universität Bielefeld): A spectral theory of regular sequences (video) (slides) (paper)

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.

March 1, 2022

Daniel Krenn (Universität Salzburg): k-regular sequences: Asymptotics and Decidability (video) (slides) (paper1) (paper2)

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.

This is based on joint works with Clemens Heuberger and with Jeffrey Shallit.

February 15, 2022

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

For pairs of real bases 𝛽_{0},𝛽_{1}>1, we study expansions of the form Σ_{k=1}^{∞} i_{k} / (𝛽_{i1} 𝛽_{i2} ... 𝛽_{ik}) with digits i_{k} ∈ {0,1}.
We characterise the pairs admitting non-trivial unique expansions as well as those admitting uncountably many unique expansions, extending recent results of Neunhäuserer (2021) and Zou, Komornik and Lu (2021). Similarly to the study of unique 𝛽-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.

February 8, 2022

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

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

February 1, 2022

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

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

January 25, 2022

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

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

January 18, 2022

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

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

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

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

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

January 11, 2022

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

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