# 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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January 31, 2023, 14:00 CET (UTC +1)

Slade Sanderson (Universiteit Utrecht): Matching for parameterised symmetric golden maps (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.

March 7, 2023, 14:00 CET (UTC +1)

Derong Kong (Chongqing University): TBA

March 21, 2023, 14:00 CET (UTC +1)

Demi Allen (University of Exeter): Diophantine Approximation for systems of linear forms - some comments on inhomogeneity, monotonicity, and primitivity (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).

## Past talks

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 D_{N} in N, a sequence ({b^{n}α})_{n≥0}, b ≥ 2 integer, can have for some real number α? If lim_{N→∞} D_{N} = 0 then α 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} ≪ 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} ≫ log^{2} N for Levin's binary normal number. So EITHER ND_N ≪ log^{2} N 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 ≪ log^{2} N is the best order for D_{N} in N a normal number can obtain.

December 13, 2022

Hiroki Takahasi (Keio University): Distribution of cycles for one-dimensional random dynamical systems (video) (slides) (paper)

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.

December 6, 2022

Christoph Bandt (Universität Greifswald): Automata generated topological spaces and self-affine tilings (video) (slides)

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.

November 29, 2022

Manuel Hauke (TU Graz): The asymptotic behaviour of Sudler products (video) (slides) (paper1) (paper2) (paper3)

Given an irrational number α, we study the asymptotic behaviour of the Sudler product defined by P_{N}(α) = Π_{r=1}^{N} 2 |sin(π r α)|, which appears in many
different areas of mathematics. In this talk, we explain the connection between the size of P_{N}(α) and the Ostrowski expansion of N with respect to α. We show that lim inf_{N→∞} P_{N}(α) = 0 and lim sup_{N→∞} P_{N}(α)/N = ∞, whenever the sequence of partial quotients in the continued fraction expansion of α exceeds 7 infinitely often, and show that the value 7 is optimal.

For Lebesgue-almost every α, we can prove more: we show that for every non-decreasing function ψ: (0,∞) → (0,∞) with ∑_{k=1}^{∞} 1/ψ(k) = ∞ and
lim inf_{k→∞} ψ(k)/(k log k) sufficiently large, the conditions log P_{N}(α) ≤ −ψ(log N), log P_{N}(α) ≥ ψ(log N) hold on sets of upper density 1 respectively 1/2.

November 22, 2022

Faustin Adiceam (Université Paris-Est Créteil): Badly approximable vectors and Littlewood-type problems (video) (slides) (paper)

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).

November 15, 2022

Seul Bee Lee (Institute for Basic Science): Regularity properties of Brjuno functions associated with by-excess, odd and even continued fractions (video) (slides) (paper)

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.

November 8, 2022

Wen Wu (South China University of Technology): From the Thue-Morse sequence to the apwenian sequences (video) (slides) (journal) (arXiv)

In this talk, we will introduce a class of ±1 sequences, called the apwenian sequences. The Hankel determinants of these ±1 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 ≥ 2 is an integer. Moreover, the number of ±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.

October 25, 2022

Álvaro Bustos-Gajardo (The Open University): Quasi-recognizability and continuous eigenvalues of torsion-free S-adic systems (video) (slides) (paper)

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.

October 18, 2022

Yufei Chen (TU Delft): Matching of orbits of certain N-expansions with a finite set of digits (video) (slides) (paper)

In this talk we consider a class of continued fraction expansions: the so-called *N-expansions with a finite digit set*, where N ≥ 2 is an integer. For N fixed, they are steered by a parameter α ∈ (0,√N−1]. For N = 2 an explicit interval [A,B] was determined, such that for all α ∈ [A,B] the entropy h(T_{α}) of the underlying Gauss-map T_{α} is equal. In this paper we show that for all integers N ≥ 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_{α}, the T_{α}-invariant measure, ergodicity, and we show that for any two α, α' 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*.

October 11, 2022

Lukas Spiegelhofer (Montanuniversität Leoben): Primes as sums of Fibonacci numbers (video) (slides) (paper)

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).

October 4, 2022

David Siukaev (Higher School of Economics): Exactness and ergodicity of certain Markovian multidimensional fraction algorithms (video) (slides)

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

Alexandra Skripchenko (Higher School of Economics): Bruin-Troubetzkoy family of interval translation mappings: a new glance (video)

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.

September 27, 2022

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

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.

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).