Abstracts
Yann Bugeaud (Univ. Strasbourg): Minicourse: On the b-ary expansion of e
Let b ≥ 2 be an integer. The exponent vb (resp., v'b) and the uniform exponent v̂b (resp., v̂'b) measure the quality of approximation to a real number by rational numbers whose denominator is a power of b (resp., is of the form br(bs-1)). Said differently and informally, we look at the lengths of the blocks of digit 0 (or of digit (b-1)) and at the lengths of repeated blocks in the base-b expansion of a real number. We discuss several results on these four exponents and explain how an inequality between v'b and v̂'b implies that the base-b expansion of any real number whose irrationality exponent is sufficiently close to 2 cannot be too 'simple', in the sense that it contains at least cn different blocks of digits of length n, for some c > 1 and every integer n sufficiently large. In particular, the b-ary expansion of e contains at least 10n/9 different blocks of digits of length n, if n is large enough.
Olivier Carton (Univ. Paris Cité, IRIF) : Rauzy dimension and finite state dimension
In a paper of 1976, Rauzy studied two complexity notions, β and β, for infinite sequences over a finite alphabet. The function β is maximum exactly in the Borel normal sequences and β is minimum exactly in the sequences that, when added to any Borel normal sequence, the result is also Borel normal. Although the definitions of β and β do not involve finite-state automata, we establish some connections between them and the lower dim and upper dim finite-state dimension (or other equivalent notions like finite-state compression ratio, aligned-entropy or cumulative log-loss of finite-state predictors). We show tight lower and upper bounds on dim and dim as functions of β and β, respectively. In particular this implies that sequences with dim zero are exactly the ones that that, when added to any Borel normal sequence, the result is also Borel normal. We also show that the finite-state dimensions dim and dim are essentially subadditive.
Bastián Espinoza (Univ. Liège) : Equicontinuous factors of S-adic subshifts
The study of dynamical systems using hierarchical decompositions has shown to be a powerful method for understanding their main properties, such as ergodic measures, automorphisms, orbit equivalence classes, and spectrum. In symbolic systems, these decompositions are usually decribed by a sequence of substitutions. This is referred to as the S-adic formalism. In this talk, I will focus on the maximal equicontinuous factor map (both topological and measure-theoretical) of symbolic systems from the S-adic point of view. I will present extensions of the known criteria for determining such factor maps and their generic fiber size to more general S-adic structures, including many with infinite alphabet rank.
Toghrul Karimov (Max Planck Institute for Software Systems): Ode to O-minimality
Let S be a mathematical structure consisting of the set of real numbers as the universe and some functions with real inputs and outputs. The structure S is o-minimal if every set definable in S has finitely many connected components. I will describe some recent applications of o-minimality theory to linear dynamical systems as well as rotations on a torus with moving targets.
Niels Langeveld (Leoben): Quadratic irrationals, (N,alpha)-continued fractions and greedy N-continued fraction expansions
A real number has a periodic regular continued fraction expansion if and only if it is a quadratic irrational. In this talk, we will look at what we can say about N-continued fractions. These are expansions of the form x = N/(d1+N/(d2+...)) where N > 1 and di ∈ N. N-continued fractions and regular continued fractions differ in many ways. For example, N-continued fractions are not unique. We will look at two different ways of generating continued fractions of the form above. The first one will generate (N,alpha)-continued fraction expansions and the second one will generate the greedy N-continued fractions. In the first case we will show that there are non-periodic quadratic irrationals for some choices of N and alpha and in the second case we highlight some numerical methods to support the hypothesis there are quadratic irrationals with a non-periodic expansion as well. This talk is joint work with Cor Kraaikamp.
Gunther Leobacher (University of Graz) and Alexander Steinicke (Montanuniversität Leoben): Permeable Sets
We present the notion of permeability of a set in Rd (and its generalization to metric spaces). This quite intuitive concept originally emerged from the study of stochastic dif- ferential equations with discontinuous coefficients and immediately turned to be studied in itself.
Permeability of a set Θ roughly means that any two points x and y of a space can be connected by a ’short curve’ which avoids hitting Θ ’too often’. Further, permeability allows to conclude global Lipschitz continuity of a continuous function, that is ’intrinsically Lipschitz’ with exception set Θ, leading to a servicable notion of ’piecewise Lipschitz continuity’ in Rd.
For subsets in Rd, in particular iterated function systems, we give a number of examples, non-examples, properties and conditions, and we illustrate that the concept of permeability leads to a rich theory in metric topology.
This is based on joint work with Jörg Thuswaldner and Tapio Rajala.
Florian Luca (Stellenbosch): On a question of Douglass and Ono
It is known that the partition function p(n) obeys Benford's law in any integer base b ≥ 2. In a recent paper, Douglass and Ono asked for an explicit version of this result. In my talk, I will show that for any string of digits of length f in base b, there is n ≤ exp(1032(f+11)2(log b)3) such that p(n) starts with the given string of f digits in base b. The proof uses a lower bound for a nonzero linear form in logarithms of algebraic numbers with algebraic coefficients due to Philippon and Waldschmidt. A similar result holds for the plane partition function.
Joris Nieuwveld (Saarbrücken): On Presburger arithmetic extended with multiple powers
In this talk, I will discuss the current state of the art of Presburger arithmetic extended with multiple powers. In particular, I will sketch a proof of the decidability of the existential fragment of ⟨Z; 0, 1, +, <, 2^N, 3^N⟩, where 2^N and 3^N denotes the set of all powers of 2 and 3, respectively. Next, I will discuss possible generalizations of this results before showing that the full theory is undecidable.