Van der Corput sets in $ \mathbb Z^d$

Volume 110 / 2008

Vitaly Bergelson, Emmanuel Lesigne Colloquium Mathematicum 110 (2008), 1-49 MSC: 11K06, 28D05, 37A45, 42A82. DOI: 10.4064/cm110-1-1


In this partly expository paper we study van der Corput sets in ${\mathbb Z}^d$, 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 $d=1$ by Kamae and M. Mendès France and by Ruzsa, 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.


  • Vitaly BergelsonDepartment of Mathematics
    The Ohio State University
    Columbus, OH 43210, U.S.A.
  • Emmanuel LesigneLaboratoire de Mathématiques
    et Physique Théorique
    (UMR CNRS 6083)
    Fédération de Recherche Denis Poisson
    Université François Rabelais
    Parc de Grandmont, 37200 Tours, France

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