A van der Corput lemma and weak mixing over groups

TL;DR

This paper investigates weak mixing properties of measure-preserving systems under abelian group actions, utilizing a van der Corput lemma for Hilbert space valued functions to analyze mixing of all orders.

Contribution

It introduces a van der Corput lemma tailored for Hilbert space valued functions on topological groups, advancing the analysis of weak mixing in dynamical systems.

Findings

Established a van der Corput lemma for Hilbert space functions on groups

Analyzed weak mixing of all orders for abelian group actions

Provided new tools for studying measure-preserving dynamical systems

Abstract

We study weak mixing of all orders for weakly mixing measure preserving dynamical systems, where the dynamics is given by the action of an abelian second countable topological group which has an invariant measure under the group operation. One of the main technical tools we use is a van der Corput lemma for Hilbert space valued functions on a second countable topological group.