Cubic ergodic averages for actions of amenable groups

TL;DR

This paper generalizes and extends results on cubic ergodic averages and positive density sets in product groups, unifying previous work on commuting actions of amenable groups.

Contribution

It provides a comprehensive generalization of earlier results on cubic ergodic averages for multiple commuting actions of amenable groups.

Findings

Unified previous results on cubic ergodic averages

Extended to multiple commuting actions of amenable groups

Generalized positive density set results in product groups

Abstract

We unify and extend some previous results about cubic ergodic averages and sets of positive density in products of groups. This provides a joint generalization of earlier work of the author in the case of two commuting actions of an amenable group, and of Q. Chu's recent result on a a finite number of commuting transformations.