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

Saúl Rodríguez Martín
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引用次数: 0

Abstract

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.
弗斯滕伯格对应原理的逆定理及其在范德尔科普特集合中的应用
在这篇文章中,我们给出了可数组中的van der Corput(vdC)集、nice vdC集和nice recurrence集(定义如下)等概念的特征。这使我们能够证明漂亮的 vdC 集是漂亮递归集,并且 vdC 集与定义它们的 F{o}lner 序列无关,从而回答了伯格森和勒格涅在可数可门群中提出的问题。我们还给出了无边群中 vdC 集的谱特征。本文所发展的方法使我们能够建立弗斯滕伯格对应原理的逆定理。此外,我们还介绍了一般非可变群中的 vdC 集,并建立了它们的一些基本性质,如分割正则性。本文中的一些结果,包括与弗斯滕伯格对应原理的逆定理,也已由罗宾-塔克-德罗布(RobinTucker-Drob)和索海尔-法汉吉(Sohail Farhangi)在他们的文章《可门群中的范德科普特集及其他》中独立证明,这篇文章将与本文同时上传到 arXiv。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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