Invited Speakers

• Veronica Becher
• De-Jun Feng
• Charlene Kalle
• Hajime Kaneko
• Jakub Konieczny
• Daniel Krenn
• Cathy Swaenepoel

Veronica Becher

Title: Normal numbers with digit dependencies

Abstract: In this talk I present metric theorems for the property of Borel normality for real  numbers under the assumption of digit dependencies in their expansion in a given integer base. We quantify precisely how much digit dependence can be allowed such that, still, almost all real numbers are  normal. Our theorem states that almost all real numbers are normal when at least slightly more than log log n consecutive digits with indices starting at position n are independent. As the main application, we consider the Toeplitz set T_P, which is the set of all sequences a₁a₂... of digits from {0,...,b-1} such that a_n is  equal to a_pn, for every p in P and n=1,2,... Here b is an integer base and P is a finite set of prime  numbers. We show that almost every real number whose base b expansion is in T_P is normal  to base b. In the case when P is the singleton set {2} we prove that more is true: almost every real number whose base b expansion is in T_P is  normal to all integer bases. This is joint work with Christoph Aistleitner and Olivier Carton. Also I present, for any given integer base and for a given set of primes P, possibly infinite (but leaving many out), an instance of a simply normal number in T_P. This is joint work with Agustín Marchionna and Gérald Tenenbaum.

De-Jun Feng

Title: Numeration systems with a delone spectrum

Abstract: Let A be a finite set of non-negative real numbers containing 0, and let q>1. Write X^A(q) for the set of numbers of the form x=a₀+a₁q+...+a_nqⁿ, n=0,1,... with a_i belonging to A. The set X^A(q) is said to be a delone set if it is uniformly discrete and relatively dense in [0,∞). In this talk, we will present some partial results on the characterization of (A,q) such that X^A(q) is a delone set. It is based on joint work with Ching-Yin Chan. 

Charlene Kalle

Title: Shifted Hurwitz complex continued fractions

Abstract: In 1887 A. Hurwitz introduced a complex continued fraction algorithm that produces for each number in the square [-1/2, 1/2] + [-1/2, 1/2]i a continued fraction expansion with digits in the set of Gaussian integers. Throughout the years several people have investigated the dynamical properties of the corresponding dynamical system. In 2017 Ei, Ito, Naked and Natsui have used the fact that this system admits a Markov partition to construct a natural extension for the system and deduce properties about the invariant measure. In this talk we consider shifted versions of the Hurwitz system and derive some of their dynamical properties. This is joint work with Fanni Sélley and Jörg Thuswaldner.

Hajime Kaneko

Title: Analogue of the Markoff-Lagrange spectrum and uniform distribution theory

Abstract: The Markoff-Lagrange spectrum, which is defined by the rational approximation property of real numbers, exhibits fascinating property. For instance, the right-end part of the spectrum, called the Hall's ray, is related to the sum set of Cantor sets. Moreover, the left-end part, called the discrete part, is related to balanced words and Markoff numbers. In this talk, we consider uniform distribution theory related to linear recurrences. Interestingly, the maximal limit points of the fractional parts of general linear recurrences have analogous property with the Markoff-Lagrange spectrum. For instance, we investigated analogy of the Hall's ray for the fractional parts of linear recurrences. Moreover, analogy of the discrete part is related to balanced words and the Thue-Morse sequence. This is a joint work with Shigeki Akiyama and Teturo Kamae.

Jakub Konieczny

Title: Asymptotically automatic sequences and Cobham's theorem

Abstract: Automatic sequences can be defined in many equivalent ways, each shedding light from a different perspective. One of the simplest definitions involves the notion of the k-kernel of a sequence, that is, the family of all sequences which can be obtained by restricting the original sequence to a residue class modulo a power of k. A sequence is k-automatic if and only if its k-kernel is finite. I introduce a new class of sequences, tentatively called "asymptotically k-automatic", which are only required to have finite k-kernels up to equality almost everywhere. This class is considerably larger than the class of k-automatic sequences, and shares many (but not all) of its interesting properties. Cobham's theorem is one of the most fundamental results in the theory of automatic sequences. It asserts, roughly speaking, that a sequence cannot be automatic with respect to two different bases k and l, except for the arguably trivial cases where k and l are multiplicatively dependent (and hence lead to the same notion of automaticity) or where the sequence is eventually periodic (and hence automatic in each base). Various extensions and analogues of this theorem have been studied. During my talk I will present a slightly weaker analogue of Cobham's theorem for asymptotically automatic sequences and show that the direct analogue is false.

Daniel Krenn

Title: k-regular sequences: computations and asymptotic analysis

Abstract: The central topic of this talk are k-regular sequences in the sense of Allouche and Shallit. These sequences have a connection to base-k systems. The formal introduction will be accompanied my some simple and some advanced examples, such as the binary sum of digits or the number of odd entries in the rows of Pascal's rhombus. We will focus on the computation of properties of k-regular sequences. In particular, we will discuss the process of asymptotic analysis in general and on our running examples. We will also see, how to perform some computations in the mathematics software SageMath. And, we will discuss properties of k-regular sequences that cannot be decided (algorithmically).

Cathy Swaenepoel

Title: Primes and squares with preassigned digits

Abstract: Bourgain (2015) estimated the number of prime numbers with a positive proportion of preassigned digits in base 2.  We first present a generalization of this result to any base g>=2.  We then discuss a more recent result for the set of squares, which may be seen as one of the most interesting sets after primes.  More precisely, for any base g>=2, we obtain an asymptotic formula for the number of squares with a proportion c>0 of preassigned digits. Moreover we provide explicit admissible values for c depending on g.  Our proof mainly follows the strategy developed by Bourgain for primes in base 2, with new difficulties for squares. It is based on the circle method and combines techniques from harmonic analysis together with arithmetic properties of squares and bounds for quadratic Weyl sums.