Multiple recurrence for two commuting transformations

TL;DR

This paper investigates the multiple recurrence properties of two commuting transformations, extending known results for single transformations using advanced machinery like magic systems.

Contribution

It introduces new results on multiple recurrence for two commuting transformations, building on and extending prior work by Bergelson, Host, and Kra.

Findings

Derived a new multiple recurrence result for two commuting transformations

Extended the theory of recurrence beyond single transformations

Utilized magic systems to prove the main results

Abstract

This paper is devoted to a study of the multiple recurrence of two commuting transformations. We derive a result which is similar but not identical to that of one single transformation established by Bergelson, Host and Kra. We will use the machinery of "magic systems" established recently by B. Host for the proof.