Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-29T02:00:33.406Z Has data issue: false hasContentIssue false
  • English
  • Français

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])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1]Auslander, L., Green, L. and Hahn, F.. Flows on Homogeneous Spaces (Annals of Mathematics Studies, 53). Princeton University Press, Princeton, NJ, 1963.CrossRefGoogle Scholar
[2]Bergelson, V., Furstenberg, H. and Weiss, B.. Piecewise-Bohr Sets of Integers and Combinatorial Number Theory (Algorithms and Combinatorics, 26). Springer, Berlin, 2006, pp. 13–37.Google Scholar
[3]Bergelson, V., Host, B. and Kra, B.. With an appendix by Imre Ruzsa. Multiple recurrence and nilsequences. Invent. Math. 160 (2005), 261–303.CrossRefGoogle Scholar
[4]Bilu, Y.. Addition of sets of integers of positive density. J. Number Theory 64 (1997), 233–275.CrossRefGoogle Scholar
[5]Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, NJ, 1981.CrossRefGoogle Scholar
[6]Gowers, W. T.. A new proof of Szemerédi’s theorem. Geom. Funct. Anal. 11 (2001), 465–588.CrossRefGoogle Scholar
[7]Green, B. and Tao, T.. Linear equations in the primes. Ann. of Math. (2) to appear.Google Scholar
[8]Host, B. and Kra, B.. Nonconventional ergodic averages and nilmanifolds. Ann. of Math. (2) 161 (2005), 397–488.CrossRefGoogle Scholar
[9]Host, B. and Kra, B.. Uniformity norms on ℓ ∞ and applications. J. Anal. to appear.Google Scholar
[10]Host, B., Kra, B. and Maass, A.. Nilsequences and a structure theorem for topological dynamical systems. Preprint, arXiv:0905.3098.Google Scholar
[11]Leibman, A.. Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold. Ergod. Th. & Dynam. Sys. 25 (2005), 113–201.Google Scholar