Disjointness for measurably distal group actions and applications

TL;DR

This paper extends the concept of quasi-disjointness to group actions, proving that measurably distal systems are quasi-disjoint from all measure-preserving systems, with applications to ergodic averages and recurrence.

Contribution

It generalizes Berg's notion of quasi-disjointness to countable group actions and establishes key disjointness properties for distal systems.

Findings

Measurably distal systems are quasi-disjoint from all measure-preserving systems.

Provides necessary and sufficient conditions for disjointness when one system is distal.

Establishes a Wiener--Wintner type theorem for countable amenable groups with distal weights.

Abstract

We generalize Berg's notion of quasi-disjointness to actions of countable groups and prove that every measurably distal system is quasi-disjoint from every measure preserving system. As a corollary we obtain easy to check necessary and sufficient conditions for two systems to be disjoint, provided one of them is measurably distal. We also obtain a Wiener--Wintner type theorem for countable amenable groups with distal weights and applications to weighted multiple ergodic averages and multiple recurrence.