Multiple recurrence for non-commuting transformations along rationally independent polynomials

Abstract

AbstractWe 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} $ 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.