Multiple recurrence for non-commuting transformations along rationally independent polynomials

TL;DR

This paper establishes a multiple recurrence theorem for measure-preserving transformations along certain bivariate polynomials, expanding understanding of recurrence phenomena in non-commuting systems.

Contribution

It introduces a novel multiple recurrence result for transformations along rationally independent polynomials in two variables, contrasting with known limitations in single-variable cases.

Findings

Proves multiple recurrence for transformations along polynomials m+p_i(n).

Shows recurrence holds even when transformations do not generate a nilpotent group.

Uses reduction to nilfactors and equidistribution on nilmanifolds in proof.

Abstract

We prove a multiple recurrence result for arbitrary measure-preserving transformations along polynomials in two variables of the form $m+p_i(n)$, with rationally independent $p_i$'s with zero constant term. This is in contrast to the single variable case, in which even double recurrence fails unless the transformations generate a virtually nilpotent group. The proof involves reduction to nilfactors and an equidistribution result on nilmanifolds.