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

Saúl Rodríguez Martín

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 Ferré Moragues \cite{BF}, Kelly and Lê \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}.

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

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 Ferré Moragues \cite{BF}, Kelly and Lê \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}.
Paper Structure (8 sections, 41 theorems, 106 equations)

This paper contains 8 sections, 41 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Some applications of the inverse correspondence principle
  4. Correspondence between Cesaro and integral averages
  5. Characterizations of vdC sets in amenable groups
  6. Spectral Characterization of vdC sets in abelian groups
  7. Properties of vdC sets
  8. Convexity and Cesaro averages

Key Result

Theorem 1.3

Let $(X,\mathcal{B},\mu,T)$ be a m.p.s. and let $C\in\mathcal{B}$ satisfy $\mu(C)>0$. Then for all $k\in\mathbb{N}$ there is some $n\in\mathbb{N}$ such that

Theorems & Definitions (104)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4: Furstenberg's correspondence principle, cf. Be5
  • Theorem 1.5
  • Theorem 1.6: Inverse Furstenberg correspondence principle
  • Remark 1.7
  • Remark 1.8
  • Definition 1.9: KM
  • Definition 1.10
  • ...and 94 more