Joint ergodicity of sequences

TL;DR

This paper establishes necessary and sufficient conditions for joint ergodicity of integer sequences, providing a flexible framework that encompasses known examples and addresses open problems without relying on deep ergodic theory tools.

Contribution

It introduces a new characterization of joint ergodicity that simplifies proofs and broadens applicability, using an ergodic variant of a combinatorial technique.

Findings

Characterization of joint ergodicity conditions

Recovery of most known jointly ergodic sequences

Resolution of some open problems in the area

Abstract

A collection of integer sequences is jointly ergodic if for every ergodic measure preserving system the multiple ergodic averages, with iterates given by this collection of sequences, converge in the mean to the product of the integrals. We give necessary and sufficient conditions for joint ergodicity that are flexible enough to recover most of the known examples of jointly ergodic sequences and also allow us to answer some related open problems. An interesting feature of our arguments is that they avoid deep tools from ergodic theory that were previously used to establish similar results. Our approach is primarily based on an ergodic variant of a technique pioneered by Peluse and Prendiville in order to give quantitative variants for the finitary version of the polynomial Szemer\'edi theorem.