Translates of S-arithmetic orbits and applications

TL;DR

This paper proves equidistribution of certain periodic orbits in S-arithmetic spaces and applies these results to improve understanding of best approximations for vectors with algebraic coordinates.

Contribution

It generalizes Eskin Mozes and Shah's work to the S-arithmetic setting, establishing new equidistribution results and applications.

Findings

Periodic orbits of the diagonal group equidistribute in S-arithmetic spaces

New insights into best approximations of algebraic vectors

Extension of measure translate equidistribution to S-arithmetic case

Abstract

We prove that certain sequences of periodic orbits of the diagonal group in the space of lattices equidistribute. As an application we obtain new information regarding the sequence of best approximations to certain vectors with algebraic coordinates. In order to prove these results we generalize the seminal work of Eskin Mozes and Shah about the equidistribution of translates of periodic measures from the real case to the S-arithmetic case.