A double return times theorem

TL;DR

This paper proves that for almost every point in a measure-preserving system, the product of two bounded functions evaluated at different iterates forms a good weight for the pointwise ergodic theorem, extending understanding of weighted ergodic averages.

Contribution

It establishes a double return times theorem showing that such products serve as good weights for pointwise ergodic convergence in measure-preserving systems.

Findings

Almost every point yields convergence of weighted averages

Products of functions at different iterates are good weights

Extends classical ergodic theorems to double return times

Abstract

We prove that for any bounded functions $f_1, f_2$ on a measure-preserving dynamical system $(X,T)$ and any distinct integers $a_1, a_2$, for almost every $x$ the sequence $$ f_1(T^{a_1 n}x) f_2(T^{a_2 n}x) $$ is a good weight for the pointwise ergodic theorem.