Spring 2022 Titles and Abstracts

Date: 3 February 2022
Speaker: Akshat Das
Title: An adelic version of the three gap theorem
Abstract: In order to understand problems in dynamics which are sensitive to arithmetic properties of return times to regions, it is desirable to generalize classical results about rotations on the circle to the setting of rotations on adelic tori. One such result is the classical three gap theorem, which is also referred to as the three distance theorem and as the Steinhaus problem. It states that, for any real number, a, and positive integer, N, the collection of points na mod 1,  where n runs from 1 to N, partitions the circle into component arcs having one of at most three distinct lengths. Since the 1950s, when this theorem was first proved independently by multiple authors, it has been reproved numerous times and generalized in many ways. One of the more recent proofs has been given by Marklof and Strömbergsson using a lattice based approach to gaps problems in Diophantine approximation. In this talk, we use an adaptation of this approach to the adeles to prove a natural generalization of the classical three gap theorem for rotations on adelic tori. This is joint work with Alan Haynes. 

Date: 10 February 2022
Speaker: Jiyoung Han
Title: The asymptotic distribution of the joint values of the integral lattice points for a system of a quadratic form and a linear form
Abstract: Let Q be a quadratic form and let L be a linear form on the n-dimensional real vector space. We are interested in the distribution of the image of the integral lattice under the map (Q, L). Developing the celebrated work of Eskin, Margulis, and Mozes in 1998, we provide the conditions of systems of forms which satisfy that the number of integral vectors in the ball of radius T whose joint values are contained in a given bounded set converges asymptotically to the volume of the region given by the level sets of the quadratic form and the linear form, intersecting with the ball of radius T, as T goes to infinity. This condition is introduced by Gorodnik in 2004.
For this, we need to classify all intermediate subgroups between the special orthogonal group preserving Q and L and the special linear group. Among them, only two closed subgroups are of our concern. We will introduce Siegel integral formulas and equidistribution theorems for each subgroup, and show how to reach our main theorem. This is joint work with Seonhee Lim and Keivan Mallahi-Karai. 

Date: 17 February 2022
Speaker: Julien Trevisan
Title: Limit laws in the lattice counting problem. The case of ellipses.
Abstract: Let E be an ellipse centered around 0. We are interested in the asymptotic distribution
of the error of the number of unimodular lattice points that fall into tE when the lattice is random
and when t goes to infinity.
Building on previous works by Bleher and by Fayad and Dolgopyat, we show that the error term, when normalized by the square root of t, converges in distribution towards an explicit distribution.
For this, we first use harmonic analysis to reduce the study of the normalized error to the study of a Siegel transform that depends on t.
Then, and this is the key part of our proof, we show that, when t goes to infinity, this last Siegel transform behaves in distribution as, what we call, a modified Siegel transform with random weights. Such objects often appear in average counting problems.
Finally, we show that this last quantity converges almost surely, and we study the existence of the moments of its law.
This work was supervised by Bassam Fayad.

Date: 3 March 2022
Speaker: Irving Calderón
Title: S-adic quadratic forms and homogeneous dynamics
Abstract: We present two new quantitative results about quadratic forms.
Let S = {\infty} \cup Sf be a finite set of places of Q. Consider the ring ZS of S-integers, and QS = \prod{p \in S} Qp. The first is a solution to the problem of deciding if any given integral quadratic forms Q1 and Q2 are ZS-equivalent. The proof is based on a reformulation of the problem in terms of the action of O(Q1, QS) on the space X{d,S} of lattices of QSd. A key tool are explicit mixing rates for the action of O(Q1, QS) on closed orbits in X{d,S}. As an application we obtain, for any S-integral orthogonal group, polynomial bounds on the S-norms of the elements of a finite generating set.
These two results and the methods of proof are based on the work of H. Li and G. Margulis for S = { \infty }.

Date: 10 March 2022
Speaker: Nattalie Tamam
Title: Classification of divergence of trajectories
Abstract: As shown by Dani, diophantine approximations are in direct correspondence to the behavior of orbits in certain homogeneous spaces. We will discuss the interpretation of the divergent trajectories and the obvious ones, the ones diverging due to a purely algebraic reason. As conjectured by Barak Weiss, there is a complete classification of divergent trajectories when considering the action of subgroups of the diagonal group. We will discuss the last part of this conjecture, showing that for a ‘large enough’ such subgroup, every divergent trajectory diverges obviously. This is a joint work with Omri Solan.

Date: 17 March 2022
Speaker: Nate Hughes
Title: Effective Counting and Spiralling of Lattice Approximates
Abstract: We will prove an effective version of Dirichlet’s approximation theorem, giving the error between the number of rational approximations to a real vector with denominator less than some real number T and the asymptotic growth of this count. Additional results for linear forms can be obtained, as well as results measuring the direction of these approximates, known as ‘spiralling of lattice approximates’. These results are obtained by reformulating the number-theoretic problem to the context of homogeneous spaces of unimodular lattices. The advantage of this reformulation is that we have more tools to deal with the problem, such as Siegel’s mean value theorem and Rogers’ higher moment formula. The proof involves using the ergodic properties of diagonal flows on this homogeneous space to calculate the number of lattice approximates, bounding the second moment of the count, then applying an effective ergodic theorem due to Gaposhkin. Particular attention is paid to the case of primitive lattices in two-dimensions, where Rogers’ theorem fails. In this case, we apply a new theorem by Kleinbock and Yu to obtain a better error term than previous results due to Schmidt.

Date: 24 March 2022
Speaker: Simon Machado
Title: Superrigidity and arithmeticity for some aperiodic subsets in higher-rank simple Lie groups
Abstract: Meyer sets are fascinating objects: they are aperiodic subsets of Euclidean spaces that nonetheless exhibit long-range aperiodic order. Sets of vertices of the Penrose tiling (P3) and Pisot-Vijarayaghavan numbers of a real number field are some of the most well-known examples. In his pioneering work, Meyer provided a powerful and elegant characterisation of Meyer sets. Years later, Lagarias proved a similar characterisation starting from what seemed to be considerably weaker assumptions.
A fascinating question asks whether Meyer’s and Lagarias’ results may be extended to more general ambient groups. In fact, a first result in that direction was already obtained in Meyer’s work: he proved a sum-product phenomenon which, implicitly, boiled down to a classification of Meyer sets in the group of affine transformations of the line.
I will talk about a generalisation of both Meyer’s and Lagarias’ theorems to discrete subsets of higher-rank simple Lie groups. I will explain how this result can be seen as a generalisation of Margulis’ arithmeticity theorem and how it can be deduced from Zimmer’s cocycle superrigidity. We will see that, surprisingly, Pisot-Vijarayaghavan numbers appear naturally in this context too. 

Date: 31 March 2022
Speaker: Johannes Schleischitz
Title: Exact uniform approximation and Dirichlet spectrum
Abstract: We consider the Dirichlet spectrum, with respect to maximum norm and simultaneous approximation. It is basically the analogue of the famous (multi-dimensional) Lagrange spectrum with respect to uniform approximation. By Dirichlet’s Theorem it is contained in [0,1]. The central new result is that it equals the entire interval [0,1] when the number of variables is two or more. We thereby get a new, constructive proof of a recent result by Beresnevich, Guan, Marnat, Ramirez and Velani that there are Dirichlet improvable vectors that are neither bad nor singular, in any dimension. We provide several generalizations, including metrical claims.

Date: 7 April 2022
Speaker: Emilio Corso
Title: Asymptotics of the equidistribution rate of expanding circles on compact hyperbolic quotients and applications
Abstract: Equidistribution properties of translates of orbits for subgroup actions on homogeneous spaces are intimately linked to the mixing features of the global action of the ambient group. The connection appears already in Margulis’ thesis (1969), displaying its full potential in the work of Eskin and McMullen during the nineties. On a quantitative level, the philosophy underpinning this linkage allows to transfer mixing rates to effective estimates for the rate of equidistribution, albeit at the cost of a sizeable loss in the exponent. In joint work with Ravotti, we instead resort to a spectral method, pioneered by Ratner in her study of quantitative mixing of geodesic and horocycle flows, in order to obtain the precise asymptotic behaviour of averages of regular observables along expanding circles on compact hyperbolic surfaces. The primary goal of the talk is to outline the salient traits of this method, illustrating how it leads to the relevant asymptotic expansion. In addition, we shall also present applications of the main result to distributional limit theorems and to quantitative error estimates on the corresponding hyperbolic lattice point counting problem, the latter having been examined, to date, only through number-theoretical methods in works of Selberg, Lax-Phillips and Phillips-Rudnick.

Date: 14 April 2022
Speaker: Mikolaj Fraczyk
Title: Thin part of the arithmetic orbifolds
Abstract: Let X be a symmetric space. The collar lemma, also known as the Margulis lemma, says that there exists an epsilon=epsilon(X), such that the epsilon-thin part of a locally symmetric space X/\Gamma looks locally like a quotient by a virtually unipotent subgroup. It turns out that in the arithmetic setting we can improve this lemma by making the epsilon grow linearly in the degree of the number filed generated by the traces of elements of \Gamma. I will explain why this is the case and present several applications, including the proof of the fact that an arithmetic locally symmetric manifold M is homotopy equivalent to a simplicial complex of size bounded linearly in the volume of M and degrees of all vertices bounded uniformly in terms of X. Based on a joint work with Sebastian Hurtado and Jean Raimbault.

Date: 28 April 2022
Speaker: Ian Hoover
Title: Effective Equidistribution on Hilbert Modular Surfaces
Subtitle: and an application to counting quadratic forms of square discriminant
Abstract: While ineffective equidistribution has been understood much more generally, effective results for non-compact orbits have been more scarce. I will give effective (polynomial) error rates for the translates of diagonal orbits on Hilbert modular surfaces. This work follows as a higher dimensional extension of the work of Kelmer and Kontorovich.

Date: 5 May 2022
Speaker: Jiajie Zheng
Title: Dynamical Borel–Cantelli Lemma for Lipschitz Twists
Abstract: In the study of some dynamical systems, the limit superior of a sequence of measurable sets is often of interest. The shrinking targets and recurrence are two of the most commonly studied problems that concern limit superior sets. However, the zero-one laws for the shrinking targets and recurrence are usually treated separately and proved differently. In this talk, we construct a generalized definition that can specialize into the shrinking targets and recurrence and our approach gives a unified proof to the zero-one laws for the two problems.