Asymptotic dynamics on amenable groups and van der Corput sets

TL;DR

This paper proves that the concept of van der Corput sets in amenable groups is independent of the choice of F4lner sequence, extending known characterizations from natural numbers to arbitrary countably infinite groups.

Contribution

It establishes the independence of van der Corput sets from F4lner sequences and generalizes their characterizations to all countably infinite groups.

Findings

Van der Corput sets are independent of F4lner sequence choice.

Many characterizations of van der Corput sets in 4mathbb{N} hold for all countably infinite groups.

A converse to the Furstenberg Correspondence Principle for amenable groups is established.

Abstract

We answer a question of Bergelson and Lesigne by showing that the notion of van der Corput set does not depend on the F\o lner sequence used to define it. This result has been discovered independently by Sa\'ul Rodr\'iguez Mart\'in. Both ours and Rodr\'iguez's proofs proceed by first establishing a converse to the Furstenberg Correspondence Principle for amenable groups. 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 F\o lner sequences remain equivalent for arbitrary countably infinite groups.