Convergence of weighted polynomial multiple ergodic averages

TL;DR

This paper investigates the convergence properties of weighted polynomial multiple ergodic averages, identifying conditions under which these averages converge in $L^2$, and extends understanding from the linear case to more general settings.

Contribution

It establishes a necessary condition for universal convergence of weighted polynomial averages and shows that for almost every point, the sequence derived from a bounded measurable function is universally good.

Findings

Identifies a necessary condition for universal convergence.

Shows that for almost every point, the sequence is universally good.

Extends results from linear to polynomial cases.

Abstract

We study here weighted polynomial multiple ergodic averages. A sequence of weights is called universally good if any polynomial multiple ergodic average with this sequence of weights converges in $L^{2}$. We find a necessary condition and show that for any bounded measurable function $\phi$ on an ergodic system, the sequence $\phi(T^{n}x)$ is universally good for almost every $x$. The linear case was understood by Host and Kra.