The Erdős–Szüsz–Turán distribution for equivariant processes

Abstract

We resolve problems posed by Kesten and Erdős–Szüsz–Turán 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.