Van der Corput sets in Z^d

TL;DR

This paper explores van der Corput sets in multidimensional integer lattices, linking harmonic analysis and dynamical systems, and extends classical results to higher dimensions with new characterizations and examples.

Contribution

It provides multidimensional generalizations of classical van der Corput set results, introduces new recurrence-based modifications, and discusses open questions in the field.

Findings

Multidimensional versions of classical van der Corput results established

New characterizations of van der Corput sets in Z^d provided

Numerous examples and open questions discussed

Abstract

In this partly expository paper we study van der Corput sets in $\Z^d$, with a focus on connections with harmonic analysis and recurrence properties of measure preserving dynamical systems. We prove multidimensional versions of some classical results obtained for $d=1$ in \cite{K-MF} and \cite{R}, establish new characterizations, introduce and discuss some modifications of van der Corput sets which correspond to various notions of recurrence, provide numerous examples and formulate some natural open questions.