On the norm convergence of nonconventional ergodic averages

Abstract

We offer a generalization of the recent result of Tao (building on earlier results of Conze and Lesigne, Furstenberg and Weiss, Zhang, Host and Kra, Frantzikinakis and Kra and Ziegler) that the nonconventional ergodic averages associated to an arbitrary number of commuting probability-preserving transformations always converge to some limit in L^2. We prove the corresponding result for a collection of commuting actions of a larger discrete Abelian group, and gives convergence that is uniform in the start-point of the averages. While Tao's proof rests on a conversion to a finitary problem, we invoke only techniques from classical ergodic theory, so giving a new proof of his result.