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Powers of sequences and convergence of ergodic averages

Published online by Cambridge University Press:  03 August 2009

N. FRANTZIKINAKIS ,
M. JOHNSON ,
E. LESIGNE  and
M. WIERDL
N. FRANTZIKINAKIS
Affiliation:
Department of Mathematics, University of Memphis, Memphis, TN 38152, USA (email: [email protected], [email protected])
M. JOHNSON
Affiliation:
Department of Mathematics & Statistics, Swarthmore College, Swarthmore, PA 19081, USA (email: [email protected])
E. LESIGNE
Affiliation:
Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 6083), Université François Rabelais Tours, Fédération de Recherche Denis Poisson, Parc de Grandmont, 37200 Tours, France (email: [email protected])
M. WIERDL
Affiliation:
Department of Mathematics, University of Memphis, Memphis, TN 38152, USA (email: [email protected], [email protected])
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Abstract

A sequence (sn) of integers is good for the mean ergodic theorem if for each invertible measure-preserving system (X,ℬ,μ,T) and any bounded measurable function f, the averages (1/N)∑ Nn=1f(Tsnx) converge in the L2(μ) norm. We construct a sequence (sn) which is good for the mean ergodic theorem but such that the sequence (s2n) is not. Furthermore, we show that for any set of bad exponents B, there is a sequence (sn) where (skn) is good for the mean ergodic theorem exactly when k is not in B. We then extend this result to multiple ergodic averages of the form (1/N)∑ Nn=1f1(Tsnx)f2(T2snx)⋯fℓ(Tℓsnx). We also prove a similar result for pointwise convergence of single ergodic averages.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 5 , October 2010 , pp. 1431 - 1456
Copyright
Copyright © Cambridge University Press 2009

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References

[1]Bergelson, V.. Weakly mixing PET. Ergod. Th. & Dynam. Sys. 7(3) (1987), 337–349.CrossRefGoogle Scholar
[2]Bergelson, V.. Combinatorial and Diophantine applications of ergodic theory, with Appendix A by A. Leibman and Appendix B by A. Quas and M. Wierdl. Handbook of Dynamical Systems, Vol. 1B. Elsevier, Amsterdam, 2006, pp. 745–869.CrossRefGoogle Scholar
[3]Bergelson, V., Host, B. and Kra, B.. Multiple recurrence and nilsequences. Invent. Math. 160 (2005), 261–303.CrossRefGoogle Scholar
[4]Boshernitzan, M., Kolesnik, G., Quas, A. and Wierdl, M.. Ergodic averaging sequences. J. Anal. Math. 95 (2005), 63–103.CrossRefGoogle Scholar
[5]Bourgain, J.. On the maximal ergodic theorem for certain subsets of the integers. Israel J. Math. 61 (1988), 39–72.CrossRefGoogle Scholar
[6]Deshouillers, J., Erdös, P. and Sárközy, A.. On additive bases. Acta Arith. 30(2) (1976), 121–132.CrossRefGoogle Scholar
[7]Deshouillers, J. and Fouvry, E.. On additive bases II. J. London Math. Soc. (2) 14(3) (1976), 413–422.CrossRefGoogle Scholar
[8]Frantzikinakis, N., Lesigne, E. and Wierdl, M.. Sets of k-recurrence but not (k+1)-recurrence. Ann. Inst. Fourier 56(4) (2006), 839–849.CrossRefGoogle Scholar
[9]Frantzikinakis, N., Lesigne, E. and Wierdl, M.. Powers of sequences and recurrence. Proc. Lond. Math. Soc. (3) 98(2) (2009), 504–530.CrossRefGoogle Scholar
[10]Host, B. and Kra, B.. Nonconventional ergodic averages and nilmanifolds. Ann. of Math. (2) 161 (2005), 397–488.CrossRefGoogle Scholar
[11]Host, B. and Kra, B.. Convergence of polynomial ergodic averages. Israel J. Math. 149 (2005), 1–19.CrossRefGoogle Scholar
[12]Host, B. and Kra, B.. Uniformity seminorms on l ∞ and applications. J. Anal. Math. to appear. Available at arXiv:0711.3637.Google Scholar
[13]Kuipers, L. and Niederreiter, H.. Uniform Distribution of Sequences (Pure and Applied Mathematics). Wiley-Interscience, New York, 1974.Google Scholar
[14]Leibman, A.. Convergence of multiple ergodic averages along polynomials of several variables. Israel J. Math. 146 (2005), 303–316.CrossRefGoogle Scholar
[15]Rosenblatt, J. and Wierdl, M.. Pointwise theorems via harmonic analysis. Ergodic Theory and its Connections with Harmonic Analysis (Alexandria, 1993) (London Mathematical Society Lecture Note Series, 205). Eds. Petersen, K. E. and Salama, I. A.. Cambridge University Press, Cambridge, 1995, pp. 3–151.CrossRefGoogle Scholar
[16]Vaughan, R.. The Hardy–Littlewood Method, 2nd edn.(Cambridge Tracts in Mathematics, 125). Cambridge University Press, Cambridge, 1997.CrossRefGoogle Scholar
[17]Ziegler, T.. Universal characteristic factors and Furstenberg averages. J. Amer. Math. Soc. 20 (2007), 53–97.CrossRefGoogle Scholar