Quantitative multiple mixing

TL;DR

This paper introduces a method to quantitatively estimate higher order correlations in group actions, proving effective mixing for various semisimple groups and applying results to lattice configurations.

Contribution

It develops a new approach for quantitative higher order mixing estimates in semisimple group actions, extending previous qualitative results.

Findings

Established effective mixing of all orders for semisimple Lie group actions

Proved effective mixing for semisimple S-algebraic and adele groups

Demonstrated existence of approximate configurations in lattices of semisimple groups

Abstract

We develop a method for providing quantitative estimates for higher order correlations of group actions. In particular, we establish effective mixing of all orders for actions of semisimple Lie groups as well as semisimple $S$-algebraic groups and semisimple adele groups. As an application, we deduce existence of approximate configurations in lattices of semisimple groups.