Multiple ergodic theorems for arithmetic sets

Abstract

We establish results with an arithmetic flavor that generalize the polynomial multidimensional Szemerédi theorem and related multiple recurrence and convergence results in ergodic theory. For instance, we show that in all these statements we can restrict the implicit parameter n n to those integers that have an even number of distinct prime factors or satisfy any other congruence condition. In order to obtain these refinements we study the limiting behavior of some closely related multiple ergodic averages with weights given by appropriately chosen multiplicative functions. These averages are then analyzed using a recent structural result for bounded multiplicative functions proved by the authors.