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

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.