Rigidity of non-maximal torus actions, unipotent quantitative recurrence, and Diophantine approximations

TL;DR

This paper introduces a new recurrence-based approach to classify positive entropy measures for higher rank diagonal actions on arithmetic quotients, with applications to Diophantine approximation.

Contribution

It develops a novel recurrence argument along unipotent directions and extends measure classification to cases without entropy assumptions.

Findings

Classification of positive entropy measures for higher rank actions on arithmetic quotients.

An adelic version of the measure classification without entropy conditions.

New results on Diophantine approximations of real numbers.

Abstract

We present a new argument in the study of positive entropy measures for higher rank diagonalisable actions. The argument relies on a quantitative form of recurrence along unipotent directions (that are not known to preserve the measure). Using this argument we prove a classification of positive entropy measures for any higher rank action on an irreducible arithmetic quotient of a form of $SL_2$. We also provide an Adelic version of this classification result where no entropy assumption is needed. These results can also be used to prove new results regarding Diophantine approximations of integer multiples of an arbitrary element $\alpha\in\mathbb{R}$.