Approximate invariance for ergodic actions of amenable groups

TL;DR

This paper introduces new techniques to analyze small doubling sets in ergodic actions of amenable groups, generalizing classical theorems and establishing optimal inverse product set results for densities.

Contribution

It develops general methods for action sets of small doubling and proves a dynamical version of Kneser's theorem for all countable amenable groups, extending classical results.

Findings

Proves a dynamical generalization of Kneser's density theorem.

Establishes new inverse product set theorems for densities.

Demonstrates the optimality of results through examples.

Abstract

We develop in this paper some general techniques to analyze action sets of small doubling for probability measure-preserving actions of amenable groups. As an application of these techniques, we prove a dynamical generalization of Kneser's celebrated density theorem for subsets in $(\bZ,+)$, valid for any countable amenable group, and we show how it can be used to establish a plethora of new inverse product set theorems for upper and lower asymptotic densities. We provide several examples demonstrating that our results are optimal for the settings under study.