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Nil–Bohr sets of integers

Published online by Cambridge University Press:  24 November 2009

BERNARD HOST  and
BRYNA KRA
BERNARD HOST
Affiliation:
Laboratoire d’analyse et de mathématiques appliquées, Université de Marne la Vallée & CNRS UMR 8050, 5 Bd. Descartes, Champs sur Marne, 77454 Marne la Vallée Cedex 2, France (email: [email protected])
BRYNA KRA
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road Evanston, IL 60208-2730, USA (email: [email protected])
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Abstract

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We study relations between subsets of integers that are large, where large can be interpreted in terms of size (such as a set of positive upper density or a set with bounded gaps) or in terms of additive structure (such as a Bohr set). Bohr sets are fundamentally abelian in nature and are linked to Fourier analysis. Recently it has become apparent that a higher order, non-abelian, Fourier analysis plays a role both in additive combinatorics and in ergodic theory. Here we introduce a higher-order version of Bohr sets and give various properties of these objects, generalizing results of Bergelson, Furstenberg, and Weiss.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 1 , February 2011 , pp. 113 - 142
Copyright
Copyright © Cambridge University Press 2009

References

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