An inverse of Furstenberg's correspondence principle and applications to van der Corput sets

TL;DR

This paper establishes an inverse of Furstenberg's correspondence principle for countable amenable semigroups, enabling new insights into recurrence sets and van der Corput sets, with spectral characterizations and foundational properties.

Contribution

It introduces an inverse correspondence principle in the setting of countable amenable semigroups, advancing understanding of recurrence and vdC sets.

Findings

Spectral characterization of vdC sets

Answering open questions on recurrence and vdC sets

Basic properties of vdC sets in amenable groups

Abstract

We obtain an inverse of Furstenberg's correspondence principle in the setting of countable cancellative, amenable semigroups. Besides being of intrinsic interest on its own, this result allows us to answer a variety of questions concerning sets of recurrence and van der Corput (vdC) sets, which were posed by Bergelson and Lesigne \cite{BL}, Bergelson and Ferr\'e Moragues \cite{BF}, Kelly and L\^e \cite{KL}, and Moreira \cite{Mor}. We also prove a spectral characterization of vdC sets and prove some of their basic properties in the context of countable amenable groups. Several results in this article were independently found by Sohail Farhangi and Robin Tucker-Drob, see \cite{FT}.