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Weighted multiple ergodic averages and correlation sequences

Published online by Cambridge University Press:  04 July 2016

NIKOS FRANTZIKINAKIS  and
BERNARD HOST
NIKOS FRANTZIKINAKIS
Affiliation:
University of Crete, Department of Mathematics, Voutes University Campus, Heraklion 71003, Greece email [email protected]
BERNARD HOST
Affiliation:
Université Paris-Est Marne-la-Vallée, Laboratoire d’analyse et de mathématiques appliquées, UMR CNRS 8050, 5 Bd Descartes, 77454 Marne la Vallée Cedex, France email [email protected]
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Abstract

We study mean convergence results for weighted multiple ergodic averages defined by commuting transformations with iterates given by integer polynomials in several variables. Roughly speaking, we prove that a bounded sequence is a good universal weight for mean convergence of such averages if and only if the average of this sequence times any nilsequence converges. Two decomposition results of independent interest play key roles in the proof. The first states that every bounded sequence in several variables satisfying some regularity conditions is a sum of a nilsequence and a sequence that has small uniformity norm (this generalizes a result of the second author and Kra); and the second states that every multiple correlation sequence in several variables is a sum of a nilsequence and a sequence that is small in uniform density (this generalizes a result of the first author). Furthermore, we use these results in order to establish mean convergence and recurrence results for a variety of sequences of dynamical and arithmetic origin and give some combinatorial implications.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 1 , February 2018 , pp. 81 - 142
Copyright
© Cambridge University Press, 2016 

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References

Assani, I., Duncan, D. and Moore, R.. Pointwise characteristic factors for Wiener–Wintner double recurrence theorem. Ergod. Th. & Dynam. Sys. to appear. Preprint, 2014, arXiv:1402.7094.Google Scholar
Assani, I. and Moore, R.. Extension of Wiener–Wintner double recurrence theorem to polynomials. J. Anal. Math. to appear. Preprint, 2014, arXiv:1409.0463.Google Scholar
Assani, I. and Moore, R.. A good universal weight for nonconventional ergodic averages in norm. Ergod. Th. & Dynam. Sys. to appear. Preprint, 2015, arXiv:1503.08863.Google Scholar
Assani, I. and Moore, R.. A good universal weight for multiple recurrence averages with commuting transformations in norm. Preprint, 2015, arXiv:1506.06730.Google Scholar
Auslander, J.. Minimal Flows and Their Extensions. North-Holland, Amsterdam, 1988.Google Scholar
Auslander, L., Green, L. and Hahn, F.. Flows on homogeneous spaces. With the assistance of L. Markus and W. Massey, and an appendix by L. Greenberg. (Annals of Mathematics Studies, 53) . Princeton University Press, Princeton, NJ, 1963.Google Scholar
Austin, T.. On the norm convergence of nonconventional ergodic averages. Ergod. Th. & Dynam. Sys. 30 (2010), 321–338.Google Scholar
Austin, T.. Pleasant extensions retaining algebraic structure, I. J. Anal. Math. 125 (2015), 1–36.CrossRefGoogle Scholar
Austin, T.. Pleasant extensions retaining algebraic structure, II. J. Anal. Math. 126 (2015), 1–111.Google Scholar
Bergelson, V.. Weakly mixing PET. Ergod. Th. & Dynam. Sys. 7(3) (1987), 337–349.Google Scholar
Bergelson, V., Host, B. and Kra, B.. Multiple recurrence and nilsequences. Invent. Math. 160(2) (2005), 261–303, With an appendix by I. Ruzsa.Google Scholar
Bergelson, V. and Leibman, A.. Polynomial extensions of van der Waerden’s and Szemerédi’s theorems. J. Amer. Math. Soc. 9 (1996), 725–753.Google Scholar
Bergelson, V. and McCutcheon, R.. An ergodic IP polynomial Szemerédi theorem. Mem. Amer. Math. Soc. 146 (2000), 695.Google Scholar
Boshernitzan, M.. An extension of Hardy’s class L of ‘Orders of Infinity’. J. Anal. Math. 39 (1981), 235–255.Google Scholar
Boshernitzan, M.. Uniform distribution and Hardy fields. J. Anal. Math. 62 (1994), 225–240.Google Scholar
Boshernitzan, M., Kolesnik, G., Quas, A. and Wierdl, M.. Ergodic averaging sequences. J. Anal. Math. 95 (2005), 63–103.Google Scholar
Camarena, A. and Szegedy, B.. Nilspaces, nilmanifolds and their morphisms. Preprint, 2012, arXiv:1009.3825v3.Google Scholar
Chu, Q.. Convergence of weighted polynomial multiple ergodic averages. Proc. Amer. Math. Soc. 137(4) (2009), 1363–1369.Google Scholar
Chu, Q., Frantzikinakis, N. and Host, B.. Commuting averages with polynomial iterates of distinct degrees. Proc. Lond. Math. Soc. (3) 102(5) (2011), 801–842.Google Scholar
Conze, J.-P. and Lesigne, E.. Théorèmes ergodiques pour des mesures diagonales. Bull. Soc. Math. France 112(2) (1984), 143–175.Google Scholar
Conze, J.-P. and Lesigne, E.. Sur un théorème ergodique pour des mesures diagonales. Probabilités, Publ. Inst. Rech. Math. Rennes, 1987-1. Université Rennes I, Rennes, 1988, pp. 1–31.Google Scholar
Daboussi, H.. Fonctions multiplicatives presque périodiques B. D’après un travail commun avec Hubert Delange. Journées Arithmétiques de Bordeaux (Conference, Université Bordeaux, Bordeaux, 1974) (Astérisque, 24–25) . Société Mathématique de France, Paris, 1975, pp. 321–324.Google Scholar
Daboussi, H. and Delange, H.. Quelques proprietes des functions multiplicatives de module au plus egal 1. C. R. Acad. Sci. Paris Ser. A 278 (1974), 657–660.Google Scholar
Daboussi, H. and Delange, H.. On multiplicative arithmetical functions whose modulus does not exceed one. J. Lond. Math. Soc. (2) 26(2) (1982), 245–264.Google Scholar
Eisner, T.. Linear sequences and weighted ergodic theorems. Abstr. Appl. Anal. (2013), Art. ID 815726.Google Scholar
Eisner, T. and Krause, B.. (Uniform) convergence of twisted ergodic averages. Ergod. Th. & Dynam. Sys. to appear. Preprint, 2014, arXiv:1407.4736.Google Scholar
Elliott, P.. Probabilistic Number Theory I. Springer, New York, 1979.Google Scholar
Frantzikinakis, N.. Equidistribution of sparse sequences on nilmanifolds. J. Anal. Math. 109 (2009), 353–395.Google Scholar
Frantzikinakis, N.. Multiple correlation sequences and nilsequences. Invent. Math. 202(2) (2015), 875–892.CrossRefGoogle Scholar
Frantzikinakis, N. and Host, B.. Higher order Fourier analysis of multiplicative functions and applications. J. Amer. Math. Soc. to appear. Preprint, 2014, arXiv:1403.0945.Google Scholar
Frantzikinakis, N. and Host, B.. Multiple ergodic theorems for arithmetic sets. Trans. Amer. Math. Soc. to appear. Preprint, 2015, arXiv:1503.07154.Google Scholar
Frantzikinakis, N., Host, B. and Kra, B.. The multidimensional Szemerédi theorem along shifted primes. Israel J. Math. 194(1) (2013), 331–348.Google Scholar
Furstenberg, H.. Recurrence in ergodic theory and combinatorial number theory. Princeton University Press, Princeton, NJ, 1981.Google Scholar
Furstenberg, H. and Weiss, B.. A mean ergodic theorem for (1/N)∑ n=1 N f (T n x)g (T n 2 x). Convergence in ergodic theory and probability (Columbus, OH, 1993) (Ohio State University Mathematical Research Institute Publications, 5) . de Gruyter, Berlin, 1996, pp. 193–227.Google Scholar
Glasner, E.. Ergodic Theory via Joinings (Mathematical Surveys Monographs, vol. 101) . American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
Granville, A. and Soundararajan, K.. Multiplicative Number Theory: The Pretentious Approach. Book manuscript in preparation.Google Scholar
Green, B. and Tao, T.. An inverse theorem for the Gowers U 3(G)-norm. Proc. Edinb. Math. Soc. 51(1) (2008), 73–153.Google Scholar
Green, B. and Tao, T.. The quantitative behaviour of polynomial orbits on nilmanifolds. Ann. of Math. (2) 175(2) (2012), 465–540.Google Scholar
Green, B. and Tao, T.. On the quantitative distribution of polynomial nilsequences-erratum. Ann. of Math. (2) 179(3) (2014), 1175–1183. arXiv:1311.6170v3.Google Scholar
Green, B., Tao, T. and Ziegler, T.. An inverse theorem for the Gowers U s+1[N]-norm. Ann. of Math. (2) 176(2) (2012), 1231–1372.Google Scholar
Griesmer, J. T.. Ergodic averages, correlation sequences, and sumsets. PhD Thesis, Ohio State University, 2009. Available at: http://rave.ohiolink.edu/etdc/view?acc_num=osu1243973834.Google Scholar
Halász, G.. Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen. Acta Math. Acad. Sci. Hung. 19 (1968), 365–403.Google Scholar
Host, B.. Ergodic seminorms for commuting transformations and applications. Studia Math. 195 (2009), 31–49.Google Scholar
Host, B. and Kra, B.. Nonconventional ergodic averages and nilmanifolds. Ann. of Math. (2) 161 (2005), 397–488.Google Scholar
Host, B. and Kra, B.. Convergence of polynomial ergodic averages. Israel J. Math. 149 (2005), 1–19.Google Scholar
Host, B. and Kra, B.. Uniformity seminorms on ℓ ∞ and applications. J. Anal. Math. 108 (2009), 219–276.Google Scholar
Host, B. and Kra, B.. A point of view on Gowers uniformity norms. New York J. Math. 18 (2012), 213–248.Google Scholar
Kátai, I.. A remark on a theorem of H. Daboussi. Acta Math. Hungar. 47 (1986), 223–225.Google Scholar
Kechris, A.. Classical descriptive set theory (Graduate Texts in Mathematics, 156) . Springer, New York, 1995.Google Scholar
Leibman, A.. Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold. Ergod. Th. & Dynam. Sys. 25(1) (2005), 201–213.Google Scholar
Leibman, A.. Pointwise convergence of ergodic averages for polynomial actions of ℤ d by translations on a nilmanifold. Ergod. Th. & Dynam. Sys. 25(1) (2005), 215–225.Google Scholar
Leibman, A.. Convergence of multiple ergodic averages along polynomials of several variables. Israel J. Math. 146 (2005), 303–316.Google Scholar
Leibman, A.. Multiple polynomial correlation 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
Leibman, A.. Correction to the paper ‘Nilsequences, null-sequences, and multiple correlation sequences’. Preprint, 2012, arXiv:1205.4004.Google Scholar
Lesigne., E.. Sur une nil-variété, les parties minimales associées à une translation sont uniquement ergodiques. Ergod. Th. & Dynam. Sys. 11(2) (1991), 379–391.Google Scholar
Meiri, D.. Generalized correlation sequences, Master’s thesis, Tel Aviv University, 1990. Available at:http://taalul.com/David/Math/ma.pdf.Google Scholar
Parry, W.. Ergodic properties of affine transformations and flows on nilmanifolds. Amer. J. Math. 91 (1969), 757–771.Google Scholar
Parry, W.. Dynamical systems on nilmanifolds. Bull. Lond. Math. Soc. 2 (1970), 37–40.Google Scholar
Rudolph, D.. Eigenfunctions of T × S and the Conze–Lesigne algebra. Ergodic Theory and its Connections with Harmonic Analysis (Alexandria, 1993). (London Mathematical Society Lecture Note Series, 205) . Cambridge University Press, Cambridge, 1995, pp. 369–432.Google Scholar
Szegedy, B.. On higher order Fourier analysis. Preprint, 2012, arXiv:1203.2260v1.Google Scholar
Tao, T.. Norm convergence of multiple ergodic averages for commuting transformations. Ergod. Th. & Dynam. Sys. 28(2) (2008), 657–688.Google Scholar
Tao, T.. Deducing the inverse theorem for the multidimensional Gowers norms from the one-dimensional version, Blog entry, https://terrytao.wordpress.com/2015/07/24.Google Scholar
Walsh, M.. Norm convergence of nilpotent ergodic averages. Ann. of Math. (2) 175(3) (2012), 1667–1688.Google Scholar
Wirsing, E.. Das asymptotische Verhalten von Summen uber multiplikative Funktionen, II. Acta Math. Acad. Sci. Hungar 18 (1967), 411–467.Google Scholar
Ziegler, T.. Universal characteristic factors and Furstenberg averages. J. Amer. Math. Soc. 20 (2007), 53–97.Google Scholar
Zorin-Kranich, P.. Norm convergence of multiple ergodic averages on amenable groups. J. Anal. Math. to appear. Preprint, 2011, arXiv:1111.7292.Google Scholar
Zorin-Kranich, P.. A uniform nilsequence Wiener–Wintner theorem for bilinear ergodic averages. Preprint, 2015, arXiv:1504.04647.Google Scholar