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

In this article we give characterizations of the notions of van der Corput (vdC) set, nice vdC set and set of nice recurrence (defined below) in countable amenable groups. This allows us to prove that nice vdC sets are sets of nice recurrence and that vdC sets are independent of the F{\o}lner sequence used to define them, answering questions from Bergelson and Lesigne in the context of countable amenable groups. We also give a spectral characterization of vdC sets in abelian groups. The methods developed in this paper allow us to establish a converse to the Furstenberg correspondence principle. In addition, we introduce vdC sets in general non amenable groups and establish some basic properties of them, such as partition regularity. Several results in this paper, including the converse to Furstenberg's correspondence principle, have also been proved independently by Robin Tucker-Drob and Sohail Farhangi in their article `Van der Corput sets in amenable groups and beyond', which is being uploaded to arXiv simultaneously to this one.