Mixing sets for rigid transformations

TL;DR

This paper constructs rigid measure-preserving transformations that exhibit mixing along specific infinite zero-density sets, addressing a question posed by Bergelson and exploring various examples and analogues.

Contribution

It provides the first known examples of rigid transformations that are mixing along infinite zero-density sets, expanding understanding of mixing behavior in ergodic theory.

Findings

Constructed rigid transformations mixing along infinite zero-density sets

Proposed Gaussian and Poisson suspensions as examples

Discussed rigid rank one transformations and group action analogues

Abstract

For any infinite zero-density integer set M, we found a rigid measure-preserving transformation mixing along M by answering Bergelson's question. Gaussian and Poisson suspensions over infinite constructions are suggested as suitable examples. We also discuss rigid rank one transformations of finite measure that mix along a given set, and some analogues for group actions.