Van der Corput's difference theorem for amenable groups and the left regular representation

TL;DR

This paper links variants of van der Corput's Difference Theorem for amenable groups to the mixing properties of their unitary representations, revealing new connections with the left regular representation and finite-dimensional subrepresentations.

Contribution

It establishes a novel connection between vdCDT variants and the ergodic hierarchy of group representations, especially for amenable groups.

Findings

One variant of vdCDT corresponds to subrepresentations of the left regular representation.

Another variant of vdCDT corresponds to the absence of finite-dimensional subrepresentations.

Applications are provided for measure-preserving actions of countably infinite abelian groups.

Abstract

We establish a connection between two variants of van der Corput's Difference Theorem (vdCDT) for countably infinite amenable groups $G$ and the ergodic hierarchy of mixing properties of a unitary representation $U$ of $G$. In particular, we show that one variant of vdCDT corresponds to subrepresentations of the left regular representation, and another variant of vdCDT corresponds to the absence of finite dimensional subrepresentations. We then obtain applications for measure preserving actions of countably infinite abelian groups.