The Erd\"os-Sz\"usz-Tur\'an distribution for equivariant processes

TL;DR

This paper advances the understanding of probabilistic Diophantine approximation by applying homogeneous dynamics to general measure-valued processes, with implications for higher-dimensional approximation and translation surface geometry.

Contribution

It generalizes classical problems to measure-valued processes in b^n and connects them to geometric and dynamical systems applications.

Findings

Resolved longstanding problems in probabilistic Diophantine approximation.

Extended the Erd
51s-Szfczs-Ture1n distribution to new settings.

Provided new insights into the distribution of point sets in higher dimensions.

Abstract

We resolve problems posed by Kesten and Erd\"os-Sz\"usz-Tur\'an on probabilistic Diophantine approximation via methods of homogeneous dynamics. Our methods allows us to generalize the problem to the setting of general measure-valued processes in $\mathbb{R}^n$, and obtain applications to the distribution of point sets which occur in higher dimensional Diophantine approximation and the geometry of translation surfaces.