Quantitative norm convergence of double ergodic averages associated with two commuting group actions

TL;DR

This paper proves $L^p$ norm convergence of double ergodic averages for measure-preserving actions of a countably infinite abelian group, providing quantitative estimates and extending understanding of multiple ergodic averages.

Contribution

It introduces an $L^p$ norm-variation estimate for double averages along orbits, establishing their convergence in $L^p$ for any finite $p$ and bounded functions.

Findings

Established $L^p$ norm convergence of double averages

Provided quantitative $L^p$ variation estimates

Reproved convergence for all finite $p$ and bounded functions

Abstract

We study double averages along orbits for measure preserving actions of $\mathbb{A}^\omega$, the direct sum of countably many copies of a finite abelian group $\mathbb{A}$. In this article we show an $L^p$ norm-variation estimate for these averages, which in particular reproves their convergence in $L^p$ for any finite $p$ and for any choice of two $L^\infty$ functions. The result is motivated by recent questions on quantifying convergence of multiple ergodic averages.