A 1965 problem due to Danzer asks whether there exists a set with finite density in Euclidean space (i.e. « not containing too many points ») intersecting any convex body of volume one. A suitable weakening of the volume constraint leads one to the (much more recent) problem of constructing dense forests. Progress towards these problems have so far involved a very wide range of areas in mathematics (including number theory, computer science, ergodic theory, geometry and harmonic analysis). After surveying some of the known results related to the Danzer Problem and to the construction of dense forests, the talk will present some new constructions.

I will provide an introduction to a collection of results known as "arithmetic equidistribution theorems" that describe the geometry of points (in an algebraic variety) with small arithmetic complexity (defined by a notion of height with good properties). I will give simple examples and mention a few fancy versions that have been useful in different settings, with applications to the study of algebraic dynamical systems. Quantitative versions also exist.

In this talk we'll discuss some sequences of rational numbers associated to iteration of algebraic functions. These include well-known sequences such as Fibonacci and other linear recurrence sequences, more rapidly growing sequences arising from orbits of points under iteration of morphisms, and rather mysterious sequences arising as algebraic degrees of iterates of birational maps. I'll introduce the techniques that arithmetic dynamicists use to tackle these problems, and focus on the many still-open questions about these sequences.

We show that for non-degenerate integer linear recurrence sequences $(u_n)_n$ with exactly two simple dominant roots, the first-order theories of $\langle \mathbb{N};

Solvency games, introduced in 2008 by Berger, Kapur, Schulman, and Vazirani, are a class of very simple one-player gambling problems in which the player's sole objective is to remain solvent, i.e., not go bankrupt. Although memoryless deterministic optimal strategies are known always to exist, in general we do not know how to compute them, nor whether they are unique and/or ultimately periodic. For an important subclass of solvency games, such questions are tightly connected to a particular class of recurrence sequences which we introduce and discuss.

In this talk, I will provide a general context where discrepancy for sequences appears, its relations with word combinatorics and symbolic dynamics. We will go through different tools such as Ostrowski numeration, bounded remainder sets, balancedness, Rauzy fractals.

Consider the triangular grid; put a lamp at each vertex, and a switch at each upwards-pointing triangle. You begin with a lone lamp at the origin, and wish to push switches (which toggle the three adjoining lamps) in such a manner that a ball of radius n around the origin is cleared. Show that you will, at one moment, have at least $\log_2(n)$ lamps on.
This simple puzzle in in fact related to three topics:
(1) models in statistical physics for glass (which do not undergo any phase transition as the temperature T goes to 0, but rather whose viscosity increases continuously as $\exp\!\left(\frac{1}{T^2}\right)$;
(2) ergodic theory, and the almost-mixing properties of symbolic dynamical systems
(3) combinatorial group theory, and more precisely the complexity of the word problem in certain groups.
I will try to explain the connections between these areas, and the main ideas behind the combinatorial statements and their application to statistical physics. This is joint work with Ivailo Hartarsky and Ivan Mitrofanov

A sequence $(u_n)$ is holonomic (or P-finite) if it satisfies a recurrence $$ a_d(n)\,u_{n+d} = a_{d-1}(n)\,u_{n+d-1} + \cdots + a_0(n)\,u_n, $$ with polynomial coefficients $a_i(n)$. In the case where the coefficients are constants, we call $(u_n)$ a C-finite sequence, and in this setting, the celebrated Skolem–Mahler–Lech theorem states that the set of zeros is a union of finitely many arithmetic progressions and a finite set. This has since been refined to give effective upper bounds on the number of arithmetic progressions and the size of the finite set *based only on the order $d$*. In the world of P-finite sequences, an analogue of the Skolem–Mahler–Lech theorem is not known in general. However, a special case is known (since 2012) for P-finite sequences where the first and last coefficients are non-zero modulo a prime $p$. In this talk, I will discuss recent/ongoing work in which we refine this result to give effective upper bounds on the number of arithmetic progressions and the size of the finite set based only on the order $d$, the degrees of the coefficients $a_i(n)$, and the prime $p$*.

A short introduction to Presburger arithmetic is given, along with a short history of results about undecidability of arithmetic theories. We mention results of Gödel, Turing, Matiyasevich, Buchi and Xiao. Building on the result of Xiao we delve into finding lower bounds for undecidability of Presburger arithmetic extended with a square predicate in a fixed number of variables. We then show decidability of Presburger arithmetic extended with polynomial predicates in a single variable, introducing simultaneous Pell equations and relative density along the way.We discuss related hard problems standing in the way of generalization.

Nearly linear recurrences are generalisations of linear recurrences and special cases of LTI systems in control theory. The Positivity Problem for nearly linear recurrences corresponds to the non-reachability problem for LTI systems. In this talk we outline a decision procedure for the Positivity Problem for nearly linear recurrences with at most two dominant characteristic roots. The main technical lemma is a sufficient condition for $\sum_n$ $|u_n|$ to be transcendental, where $(u_n)$ is a linear recurrence. This is joint work with Amaury Pouly and Mahsa Shirmohammadi.

After the Hartmanis–Stearns conjecture, the transcendence of numbers with $\beta$-expansions in algebraic bases $\beta$ related to morphic sequences has been studied. In 2024, Pavol Kebis, Florian Luca, Joël Ouaknine, Andrew Scoones, and James Worrell proved that such expansions yield transcendental numbers or lie in $\mathbb{Q}(\beta)$ when given by so-called echoing sequences. In 2025, they proved such expansions yield transcendental numbers when the echoing sequences hold strong echoing condition. They also showed that Arnoux–Rauzy sequences are echoing and $k$-bonacci sequences are strong echoing. We extend these results to $S$-adic sequences such that the associated $S$-adic shift has purely discrete spectrum and the condition PRICE of Berthé, Steiner and Thuswaldner (2019) holds. Furthermore, we show that when such $S$-adic shift is dendric, the sequence in the $S$-adic shift is strong echoing. Our approach builds upon the geometrical interpretation of these sequences through their Rauzy fractals. This is joint work with Wolfgang Steiner.

The study of zeros of exponential polynomials lies at the heart of Skolem's Problem, which concerns the remarkably rigid structure of the zero sets of linear recurrence sequences. We study broad generalisations of exponential-polynomial equations over the field of algebraic numbers, subject to multiplicative perturbations of controlled logarithmic height. The resulting quantitative bounds depend only on logarithmic heights and require no fixed underlying number field. In the order-three setting, quantitative finiteness results are obtained for perturbative equations whose coefficients are allowed logarithmic height growing linearly in the parameter. As applications, we develop a pseudo-effective theory for Skolem's Problem, deriving explicit quantitative bounds for the zeros of non-degenerate order-three linear recurrence sequences of bounded height, up to finitely many exceptional families. The resulting thresholds for the location of the zeros, as well as the corresponding bounds on their number, are comparatively small. This perturbative framework is further extended to higher-order recurrence sequences satisfying suitable low-height root clustering conditions. The theory is also applied to a moving-target problem in linear dynamics, establishing finiteness statements for linear orbits intersecting expanding arithmetic neighbourhoods of proper subspaces.

A Carmichael number is a composite positive integer $N$ which behaves like a prime number with respect to Fermat’s Little theorem; that is $a^N - a$ is a multiple of $N$ for all integers $a$. It is known that there are infinitely many such numbers. In this talk, we will explore such numbers which have the form $2^n k + 1$ for some odd integer $k$. The questions we can ask are. Fix $k$. What can we say about the numbers $n$ such that $2^n k + 1$ is Carmichael? Or, what can we say about the odd positive integers $k$ such that $N = 2^n k + 1$ is a Carmichael number for some positive integer $n$? We will give some partial answers to these questions.