This online seminar is devoted to recent progress in Diophantine approximation, homogeneous dynamics and related topics. Every lecture will consist of two parts: the first part for a general introduction and the second part for explaining proofs in detail.

Talks will be given on zoom. Please send an email to one of the organizers to get the access.

If you would like to give a talk, please send the title, abstract and related papers (if available) of your proposed talk to one of the organizers by email.

Organizers: Victor Beresnevich, Erez Nesharim, Lei Yang

All Talks

### Title: Density of rational points near/on compact manifolds with certain curvature conditions

### Title: Linear repetitivity in polytopal cut and project sets

### Title: Generalization of Selberg's 3⁄16 theorem for convex cocompact thin subgroups of SO(n, 1)

### Title: Random matrix products and self-projective sets

Speaker: Damaris Schindler (University of Göttingen)

Date: 12 May, 2021 13:30 UTC

Abstract: In this talk I will discuss joint work with Shuntaro Yamagishi where we establish an asymptotic formula for the number of rational points, with bounded denominators, within a given distance to a compact submanifold $M$ of $\mathbb{R}^n$ with a certain curvature condition. Technically we build on work of Huang on the density of rational points near hypersurfaces. One of our goals is to explore generalisations to higher codimension. In particular we show that assuming certain curvature conditions in codimension at least two, leads to upper bounds for the number of rational points on M which are even stronger than what would be predicted by the analogue of Serre’s dimension growth conjecture.

Full InformationSpeaker: Henna Koivusalo (University of Bristol)

Date: 5 May, 2021 13:30 UTC

Abstract: Cut and project sets are aperiodic point patterns obtained by projecting an irrational slice of the integer lattice to a subspace. One way of classifying aperiodic sets is to study the number and repetition of finite patterns. From this perspective, sets with patterns repeating linearly often, called linearly repetitive sets, can be viewed as the most ordered aperiodic sets. Repetitivity of a cut and project set depends on the slope and shape of the irrational slice. The cross-section of the slice is known as the window. In earlier works, joint with subsets of {Haynes, Julien, Sadun, Walton}, we showed that many properties of cut and project sets with a cube window can be studied in the language of Diophantine approximation. For example, linear repetitivity holds if and only if the following two conditions are satisfied: (i) the cut and project set has minimal number of different finite patterns (minimal... Full Information

Speaker: Pratyush Sarkar (Yale University)

Date: 28 April, 2021 13:30 UTC

Abstract:v Selberg’s 3/16 theorem for congruence covers of the modular surface is a beautiful theorem which has a natural dynamical interpretation as uniform exponential mixing. Bourgain-Gamburd-Sarnak’s breakthrough works initiated many recent developments to generalize Selberg’s theorem for infinite volume hyperbolic manifolds. One such result is by Oh-Winter establishing uniform exponential mixing for convex cocompact hyperbolic surfaces. These are not only interesting in and of itself but can also be used for a wide range of applications including uniform resonance free regions for the resolvent of the Laplacian, affine sieve, and prime geodesic theorems. I will present a further generalization to higher dimensions and some of these immediate consequences.

Full InformationSpeaker: Natalia Jurga (University of St Andrews)

Date: 21 April, 2021 13:30 UTC

Abstract: A finite set of matrices $A \subset \mathrm{SL}(2,\mathbb{R})$ acts on one-dimensional real projective space $\mathbb{P}^1$ through its linear action on $\mathbb{R}^2$. In this talk we will be interested in the smallest closed subset of $\mathbb{P}^1$ which contains all attracting and neutral fixed points of matrices in $A$ and which is invariant under the projective action of $A$. Recently, Solomyak and Takahashi proved that if $A$ is uniformly hyperbolic and satisfies a Diophantine property, then the invariant set has Hausdorff dimension equal to the minimum of 1 and the critical exponent. In this talk we will discuss an extension of their result beyond the uniformly hyperbolic setting. This is based on joint work with Argyrios Christodoulou.

Full Information