Asymptotic dynamics on amenable groups and van der Corput sets

Abstract

We answer a question of Bergelson and Lesigne as well as a question of Fish and Skinner. The first question is answered by showing that the notion of the van der Corput set does not depend on the Følner sequence used to define it. This result has been discovered independently by Saúl Rodríguez Martín. Both ours and Rodríguez’s proofs proceed by first establishing a converse to the Furstenberg correspondence principle for amenable groups, which answers the second question. This involves studying the distributions of Reiter sequences over congruent sequences of tilings of the group. Lastly, we show that many of the equivalent characterizations of van der Corput sets in $\mathbb{N}$ that do not involve Følner sequences remain equivalent for arbitrary countably infinite groups.