Titles and Abstracts

LMS Regional Meeting

Oliver Jenkinson – Optimal properties of real numbers and their digit expansions
Every rational number has a periodic (or eventually periodic) decimal digit expansion, and this can be associated with a periodic orbit for the mapping $x \mapsto 10x\; ({\rm mod}\; 1)$ . Fixing the arithmetic mean of such an orbit, or equivalently fixing the mean value of the digits in the decimal expansion, we shall explore questions such as: Which numbers give smallest variance around the mean? Which numbers give largest geometric mean?

Donald Robertson

Tanja Schindler – Dynamical spectrum of power-free integers in quadratic number fields and beyond
Imagine you are standing in the origin of two-dimensional lattice and on each $(x,y)$ with $x,y$ integers there is an obstacle (of radius zero) blocking the sight to points behind. The points that will remain visible to you (called visible lattice points) are exactly those points, where x and y are coprime. Using a translation (i.e. going in each step a fixed distance in a fixed direction) they form a topological dynamical system with interesting properties. One can consider the square-free numbers (i.e. the integers which don’t have a prime factor of power 2 or higher) as one-dimensional analog to those systems. In particular, these systems have pure point spectrum, i.e. the spectrum consists of only the eigenvalue of the translation.
By the Halmos-von Neumann theorem two such systems are measure-theoretically isomorphic if and only if they have the same spectrum, thus establishing one of the first cases of a complete invariant in dynamical systems theory. The point of this talk will be that large and interesting families of number-theoretic dynamical systems exist where it is actually possible to determine the spectrum precisely, thus giving an explicit group-theoretic invariant to tell them apart with respect to measure-theoretic conjugacy. The talk is based on joint work with Michael Baake and Daniel Luz.

Workshop Invited Speakers

Timothee Bernard

Victor Beresnevich

Bence Borda – Continued fraction convergents under congruence conditions
One of the first interactions between dynamical systems and number theory was the observation that the ergodic and mixing properties of the Gauss map lead to probabilistic limit theorems for various statistics of the continued fraction convergents to a random real number. In this talk, we discuss a skew product extension of the Gauss map that encodes the modulo m remainders of the numerator and denominator of the convergents, and prove that the modulo m remainders satisfy the central limit theorem and the law of the iterated logarithm. The approach also has direct applications to probabilistic limit theorems for circle rotations.

Thomas Jordan

Jungwon Lee – Limit theorems for cuspidal orbits on Fuchsian groups
We discuss the statistical aspects of the quantum modular form, which was introduced by Zagier in terms of a near-modularity property on the set of rational cusps. Examples include classical modular symbols and central values of additive twists of rank 2 $L$ -functions. We outline the main ideas for the proof of their limit laws coming from dynamics and hyperbolic geometry (joint with Sandro Bettin and Sary Drappeau).

Joel Moreira

Damaris Schindler – Apollonian Circle Packings
In this talk, we discuss number-theoretic questions related to Apollonian circle packings. After defining Apollonian circle packings, we focus in particular on the sequence of curvatures that appear in such a configuration. What can we say about the occurring curvatures? Which integers appear as curvatures in a given Apollonian circle packing? To approach this question, we discuss connections to quadratic forms as well as local-global principles.

Noy Soffer Aranov – Optimal Variance Bounds for Lattice Point Counting and Quantitative Oppenheim Conjecture in Function Fields
Oppenheim conjectured that every non-degenerate indefinite irrational quadratic form $Q:\mathbb{R}^n\rightarrow \mathbb{R}$ satisfies $\overline{Q(\mathbb{Z}^n)}=\mathbb{R}$ . Oppenheim’s conjecture was eventually proved by Margulis, and later Ratner through homogeneous dynamics and measure rigidity. An interesting question pertains to the rate at which $Q(\mathbb{Z}^n)$ becomes dense, and there have been many works in this direction, such as Eskin-Margulis-Mozes, Athreya-Margulis, Ghosh-Kelmer-Yu and Kelmer-Yu. Surprisingly, in the function field setting, Oppenheim’s conjecture holds due to a result from Amir Mohammadi’s thesis, despite the fact that Ratner’s orbit closure theorem in this setting is still wide open.
In this talk, we discuss ongoing work with Jiyoung Han about quantitative density of quadratic forms defined over function fields in odd positive characteristic. To obtain our results on effective density, we compute the moments of the Siegel transform, establish a function field analogue of Rogers’ second moment formula, and apply it to obtain bounds on the discrepancy between the number of lattice points in a set and the expected value obtained through basic counting heuristics. Furthermore, we obtain an optimal bound for the variance for the number of lattice points in a set, which is obtained for indicators on balls. 

Jimmy Tseng – $\delta$ -Badly approximable numbers and ubiquitously losing sets with a view towards ergodic theory
There is a well-known method, introduced by Wolfgang Schmidt in the 1960s, for studying the set of badly approximable numbers Bad and related sets in Diophantine approximation. This method involves certain two-player games, now referred to as Schmidt games, and enables us to show that these sets of interest satisfy a countable intersection property and have full Hausdorff dimension, despite being Lebesgue-null sets. This method has since been extended to sets of points from exceptional orbits coming from certain ergodic dynamical systems.
In this talk, I will discuss an extension of this method by introducing the notion of ubiquitously losing sets and will use this notion to give upper bounds for Hausdorff dimension that are strictly less than full Hausdorff dimension. This extension enables this method to study not only sets that satisfy a countable intersection property and have full Hausdorff dimension but also sets that satisfy a finite intersection property but need not have full Hausdorff dimension. We will apply this extension to the study of the filtration of Bad by the sets of $\delta$ -badly approximable numbers, sets which arise naturally from the seminal result in Diophantine approximation, namely Dirichlet’s approximation theorem.

Pankaj Vishe – Inhomogeneous $p$ -adic Littlewood conjecture
Let $p$ be an arbitrary but fixed prime. We prove that for almost every real $\alpha$ , the following statement holds for every real $b$ and every $p$ -adic integer $c$ :
$\liminf_{|q|\rightarrow\infty} |q|\langle q\alpha+b\rangle |q+c|_p=0$
Here $\langle\cdot\rangle$ denotes distance to the nearest integer. We thus prove a $p$ -adic version of a question by Cassels. We also prove above statement for every quadratic irrational $\alpha$ . The proof is via dynamics on a suitable homogenous space. This is a joint work with Menny Akka (ETH, Zurich), Alex Gorodnik (U Zurich) and Yuval Yifrach (U Zurich).

Workshop Contributed Speakers

Dahmah Alssayyary

Connor O’Reilly – Quantitative results on the k-dimensional Duffin-Schaeffer conjecture
We discuss quantitative forms of the Duffin-Schaeffer conjecture in metric Diophantine approximation. Briefly, the conjecture concerns the existence of infinitely many efficient rational approximations to a typical irrational point with distance bounded by a given function; the quantitative version asks for the count of such approximations with bounded denominator. Recent work of Koukoulopoulos, Maynard and Yang provides an almost-sharp asymptotic for this count in the 1-dimensional case; we show that this asymptotic also holds in all higher dimensions.

Vicente Saavedra Araya

Charlie Wilson – Dynamical and statistical properties of the map $\xi \alpha^n\; {\rm mod}\; 1$ for $\alpha$ fixed
The sequence $x^n$ modulo 1 for n in the naturals has been one of immense interest for many years and at the turn of the last century there were many strong results from Hardy, Pisot, Weyl and others giving strong statements about the structure of these sequences. Despite this there are many elementary questions whose answers are unknown. For instance does there exist a transcendental $x$ whose fractional part limits to 0? We go about proving some statistical and dynamical properties of the associated map $\xi \rightarrow \xi \alpha^n$ (where $\alpha$ is fixed). For instance we examine the strong Borel Cantelli property, Shrinking target problems, recurrence problems, return time problems and Eventually always hitting problems.

Yunlong Xu – Dimensions of orbital sets in complex dynamics
We investigate the box dimension of orbital sets in complex dynamics. In particular, we are interested in the relationship between the box dimension of the orbital set and the box dimension of the associated Julia set for any rational map with degree at least 2 defined on Riemann sphere. Joint work with Jonathan Fraser.