Distribution of orbits of unipotent groups on S-arithmetic homogeneous spaces

TL;DR

This paper extends a theorem on the convergence of ergodic averages for unipotent flows to the S-arithmetic setting, providing new insights into the distribution of orbits on S-arithmetic homogeneous spaces.

Contribution

It introduces an S-arithmetic version of Dani-Margulis theorem, analyzing orbit distribution and ergodic averages in this broader context.

Findings

Proves S-arithmetic analogue of Dani-Margulis theorem

Establishes convergence of ergodic averages outside singular sets

Provides new tools for studying unipotent flows in S-arithmetic spaces

Abstract

We will prove an S-arithmetic version of a theorem of Dani-Margulis on the convergence of ergodic averages of a given bounded continuous function, when the initial point is outside certain compact subsets of the singular set associated to the unipotent flow.