{"id":107,"date":"2026-06-01T10:20:45","date_gmt":"2026-06-01T10:20:45","guid":{"rendered":"https:\/\/sites.exeter.ac.uk\/lms26\/?page_id=107"},"modified":"2026-06-25T09:07:30","modified_gmt":"2026-06-25T09:07:30","slug":"titles-and-abstracts","status":"publish","type":"page","link":"https:\/\/sites.exeter.ac.uk\/lms26\/titles-and-abstracts\/","title":{"rendered":"Titles and Abstracts"},"content":{"rendered":"\n

LMS Regional Meeting<\/h2>\n\n\n\n

Oliver Jenkinson – Optimal properties of real numbers and their digit expansions<\/strong>
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. 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?<\/p>\n\n\n\n

Donald Robertson – Patterns in Sets of Positive Density<\/strong>
The presence of an arithmetic pattern in a set of natural numbers can be expressed dynamically in terms translates of the set. If the set is large in a suitable sense, then an invariant measure can be introduced and the tools of ergodic theory applied. Which arithmetic patterns can those tools produce? Which dynamical systems arise? Answering these questions will reveal connections to number theory and homogeneous dynamics.<\/p>\n\n\n\n

Tanja Schindler – Dynamical spectrum of power-free integers in quadratic number fields and beyond<\/strong>
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.<\/p>\n\n\n\n

Workshop Invited Speakers<\/h2>\n\n\n\n

Timothee Bernard – Khintchine theorem on fractals and multislicing
<\/strong>The Khintchine theorem is one of the cornerstones of Diophantine approximation. It predicts how well Lebesgue-typical real numbers can be approximated by rationals. I will explain how to extend this theorem to typical points chosen by a fractal measure (e.g. the middle-thirds Cantor measure). This result answers a question of K. Mahler from the 1980s regarding Diophantine approximation on fractals. The proof relies on the effective equidistribution of an associated random walk on the homogeneous space \"\mathrm{SL}_{d+1}(\mathbb{R})\/\mathrm{SL}_{d+1}(\mathbb{Z})\", which in turn exploits a multislicing extension of Bourgain’s projection theorem. Joint work with Weikun He and Han Zhang.<\/p>\n\n\n\n

Victor Beresnevich<\/strong><\/p>\n\n\n\n

Bence Borda – Continued fraction convergents under congruence conditions<\/strong>
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.<\/p>\n\n\n\n

Thomas Jordan – Shrinking targets for self-affine systems
<\/strong>The shrinking target looks at a dynamical system and the set of points where the orbit hits some shrinking target infinitely often. We will look at the problem of looking at the Hausdorff dimension of such sets in certain specific situations. We will start be describing the, well understood, case of self-similar systems. We will then look at self-affine systems in the plane and where the shrinking targets are balls with a fixed centre and exponentially shrinking radius. We will show a general upper bound in this case and then look at specific example (with diagonal maps) and show that there are situations when the natural upper bound gives the dimension of the set and some situations where it does not, depending on the centre of the targets, This is joint work with Henna Koivusalo.<\/p>\n\n\n\n

Jungwon Lee – Limit theorems for cuspidal orbits on Fuchsian groups<\/strong>
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\u202fand\u202fcentral values of additive twists of rank 2 \"L\"-functions. We outline the main ideas for the proof of their limit laws coming from dynamics\u202fand\u202fhyperbolic geometry (joint with Sandro Bettin\u202fand\u202fSary Drappeau).<\/p>\n\n\n\n

Joel Moreira – The Pythagorean triples problem<\/strong>
A problem posed by Graham and Erdos asks if the Pythagorean equation \"x^2+y^2=z^2\" is partition regular (i.e., has a monochromatic solution in any finite colouring of the natural numbers). While this problem remains open, a promising approach based on ergodic theoretic ideas has led to some partial results, including a proof that in any finite colouring of the natural numbers there exist Pythagorean triples with two of its terms in the same colour. In this talk, after briefly presenting the history of this and related problems, I will outline the ideas of the proof. The talk is based on joint work with Frantzikinakis and Klurman.<\/p>\n\n\n\n

Damaris Schindler – Apollonian Circle Packings<\/strong>
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.<\/p>\n\n\n\n

Noy Soffer Aranov – Optimal Variance Bounds for Lattice Point Counting and Quantitative Oppenheim Conjecture in Function Fields<\/strong><\/strong>
Oppenheim conjectured that every non-degenerate indefinite irrational quadratic form \"Q:\mathbb{R}^n\rightarrow 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,\u202fwhich is obtained for indicators on balls.\u202f <\/p>\n\n\n\n

Jimmy Tseng – \"\delta\"-Badly approximable numbers and ubiquitously losing sets with a view towards ergodic theory<\/strong>
There is a well-known method, introduced by Wolfgang Schmidt in the 1960s, for studying the set of badly approximable numbers Bad<\/strong> 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<\/strong> by the sets of \"\delta\"-badly approximable numbers, sets which arise naturally from the seminal result in Diophantine approximation, namely Dirichlet\u2019s approximation theorem.<\/p>\n\n\n\n

Pankaj Vishe – Inhomogeneous \"p\"-adic Littlewood conjecture<\/strong>
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}
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).<\/p>\n\n\n\n

Workshop Contributed Speakers<\/h2>\n\n\n\n

Dahmah Alssayyary – Hausdorff dimension of shrinking target sets for the uniformly expanding map on the torus
<\/strong>In this talk, we obtain results on the Hausdorff dimension of limsup sets associated to a shrinking target sequence. Our results focus on geometric target shapes beyond balls, such as tubes. Our targets are generated via dynamical systems, including uniformly expanding maps on the torus. We obtain a relationship between the Hausdorff dimension and the exponential rate of shrinking of the target. This is joint work with D. Allen and M. Holland.<\/p>\n\n\n\n

Richard Howat – Affine Thickness
<\/strong>A new notion of thickness for subsets of \"B[0,1]\subset called affine thickness is defined; this notion of thickness is a generalisation of Falconer-Yavicoli thickness and is adapted to be used in the study of certain sets with affine cut outs. Thick sets can be shown to be winning for the matrix potential game introduced by Howat, Mitchell and Samuel and as an application we can demonstrate strong pattern and intersection results for thick sets. We then apply these techniques to Bedford-McMullen carpets to find examples with large patterns.<\/p>\n\n\n\n

Connor O’Reilly – Quantitative results on the k-dimensional Duffin-Schaeffer conjecture
<\/strong>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.<\/p>\n\n\n\n

Alden Paige – Dynamics of Integer Zeroes to Quadratic Forms
<\/strong>It is known that one may represent every primitive Pythagorean triple as some finite product of matrices right-multiplied by a vector, where we only have 3 choices for each matrix and 2 for the vector. This can also be thought of as giving a \u201ctree structure\u201d to the triples. In \u201cThe Dynamics of Pythagorean Triples\u201d, Romik introduced a dynamical system which reflects this tree structure (or in other words, gives information on the matrix representation in some manner). In this talk, we examine how we might approach repeating this process for other primitive integer zeroes to other quadratic forms.<\/p>\n\n\n\n

Vicente Saavedra Araya – Uniform distribution of Integer Cantor Sets
<\/strong>An integer Cantor set is a collection of integers defined by forbidding specific digits in a fixed base expansion. In this talk, I will present classical results on when such sets are uniformly distributed in congruence classes, and a recent extension obtained by connecting the problem to the behaviour of certain Markov chains. In addition, motivated by ideas from dynamical systems, I will present some extensions to integers with more general digit restrictions, such as forbidding specific combinations of digits.<\/p>\n\n\n\n

Charlie Wilson – Dynamical and statistical properties of the map \"\xi for \"\alpha\" fixed<\/strong>
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 (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.<\/p>\n\n\n\n

Yunlong Xu – Dimensions of orbital sets in complex dynamics<\/strong>
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.<\/p>\n","protected":false},"excerpt":{"rendered":"

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