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Integer part polynomial correlation sequences

Published online by Cambridge University Press:  19 September 2016

ANDREAS KOUTSOGIANNIS
ANDREAS KOUTSOGIANNIS*
Affiliation:
The Ohio State University, Department of Mathematics, Columbus, OH, USA email [email protected]
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Abstract

Following an approach presented by Frantzikinakis [Multiple correlation sequences and nilsequences. Invent. Math. 202(2) (2015), 875–892], we prove that any multiple correlation sequence defined by invertible measure preserving actions of commuting transformations with integer part polynomial iterates is the sum of a nilsequence and an error term, which is small in uniform density. As an intermediate result, we show that multiple ergodic averages with iterates given by the integer part of real-valued polynomials converge in the mean. Also, we show that under certain assumptions the limit is zero. A transference principle, communicated to us by M. Wierdl, plays an important role in our arguments by allowing us to deduce results for $\mathbb{Z}$-actions from results for flows.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 4 , June 2018 , pp. 1525 - 1542
Copyright
© Cambridge University Press, 2016 

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References

Bergelson, V.. Weakly mixing PET. Ergod. Th. & Dynam. Sys. 7(3) (1987), 337–349.Google Scholar
Bergelson, V., Host, B., Kra, B. and Ruzsa, with an appendix by I.. Multiple recurrence and nilsequences. Invent. Math. 160(2) (2005), 261–303.Google Scholar
Bergelson, V. and Leibman, A.. Distribution of values of bounded generalized polynomials. Acta Math. 198 (2007), 155–230.Google Scholar
Bergelson, V. and McCucheon, R.. Idempotent ultrafilters, multiple weak mixing and Szemerédi’s theorem for generalized polynomials. J. Anal. Math. 111 (2010), 77–130.CrossRefGoogle Scholar
Boshernitzan, M., Jones, R. L. and Wierdl, M.. Integer and fractional parts of good averaging sequences in ergodic theory. Convergence in Ergodic Theory and Probability. Eds. Bergelson, V., March, P. and Rosenblatt, J.. Walter de Gruyter, Berlin, New York, 1996, pp. 117–132.CrossRefGoogle Scholar
Chu, Q., Frantzikinakis, N. and Host, B.. Ergodic averages of commuting transformations with distinct degree polynomial iterates. Proc. Lond. Math. Soc. (3) 102 (2011), 801–842.CrossRefGoogle Scholar
Frantzikinakis, N.. Multiple correlation sequences and nilsequences. Invent. Math. 202(2) (2015), 875–892.Google Scholar
Furstenberg, H., Katznelson, Y. and Ornstein, D.. The ergodic theoretical proof of Szemerédi’s theorem. Bull. Amer. Math. Soc. (N.S.) 7 (1982), 527–552.CrossRefGoogle Scholar
Host, B. and Kra, B.. Nonconventional ergodic averages and nilmanifolds. Ann. of Math. (2) 161(1) (2005), 397–488.Google Scholar
Host, B. and Kra, B.. Uniformity seminorms on l ∞ and applications. J. Anal. Math. 108 (2009), 219–276.Google Scholar
Leibman, A.. Multiple polynomial sequences and nilsequences. Ergod. Th. & Dynam. Sys. 30(3) (2010), 841–854.Google Scholar
Leibman, A.. Nilsequences, null-sequences, and multiple correlation sequences. Ergod. Th. & Dynam. Sys. 35(1) (2015), 176–191.Google Scholar
Lesigne, E.. On the sequence of integer parts of a good sequence for the ergodic theorem. Comment. Math. Univ. Carolin. 36(4) (1995), 737–743.Google Scholar
Walsh, M.. Norm convergence of nilpotent ergodic averages. Ann. of Math. (2) 175(3) (2012), 1667–1688.Google Scholar
Wierdl, M.. Personal communication, 2015.Google Scholar