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IP-sets and polynomial recurrence

Published online by Cambridge University Press:  14 October 2010

Vitaly Bergelson
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, Ohio, 43210, USA
Hillel Furstenberg
Affiliation:
Landau Center for Analysis, Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
Randall McCutcheon
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, Ohio, 43210, USA

Abstract

We combine recurrence properties of polynomials and IP-sets and show that polynomials evaluated along IP-sequences also give rise to Poincaré sets for measure-preserving systems, that is, sets of integers along which the analogue of the Poincaré recurrence theorem holds. This is done by applying to measure-preserving transformations a limit theorem for products of appropriate powers of a commuting family of unitary operators.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

[F1]Furstenberg, H.. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. d'Analyse Math. 31 (1977), 204256.CrossRefGoogle Scholar
[F2]Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, 1981.CrossRefGoogle Scholar
[FK]Furstenberg, H. and Katznelson, Y.. An ergodic Szemerédi theorem for IP-systems and combinatorial theory. J. d'Analyse Math. 45 (1985), 117168.CrossRefGoogle Scholar
[H]Hindman, N.. Finite sums from sequences within cells of a partition of N. J. Combinat. Th. (Series A) 17 (1974), 111.Google Scholar
[M]Milliken, K.. Ramsey's theorem with sums or unions. J. Combinat. Th. (Series A) 18 (1975), 276290.CrossRefGoogle Scholar
[S]Sárközy, A.. On difference sets of integers III. Acta. Math. Acad. Sci. Hungar. 31 (1978), 125149.CrossRefGoogle Scholar
[T]Taylor, A.. A canonical partition relation for finite subsets of w. J. Combinat. Th. (Series A) 21 (1976), 137146.CrossRefGoogle Scholar