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

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}.