A good universal weight for multiple recurrence averages with commuting transformations in norm

TL;DR

This paper demonstrates that certain sequences serve as universal weights for multiple recurrence averages with commuting transformations in norm, extending key convergence results in ergodic theory.

Contribution

It introduces a new class of universal weights for multiple recurrence averages with commuting transformations, broadening the scope of Bourgain's and Tao's convergence results.

Findings

Sequences are good universal weights for multiple recurrence averages in norm.

Extends Bourgain's pointwise convergence to a broader setting.

Builds on Tao's norm convergence and previous single transformation results.

Abstract

We will show that the sequences appearing in Bourgain's double recurrence result are good universal weights to the multiple recurrence averages with commuting measure-preserving transformations in norm. This will extend the pointwise converge result of Bourgain, the norm convergence result of Tao, and the authors' previous work on the single measure-preserving transformation.