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Nilsystems and ergodic averages along primes

Part of: Topological dynamics Sequences and sets Ergodic theory

Published online by Cambridge University Press:  11 April 2019

TANJA EISNER
TANJA EISNER*
Affiliation:
Institute of Mathematics, University of Leipzig, P.O. Box 100 920, 04009Leipzig, Germany email [email protected]
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Abstract

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A celebrated result by Bourgain and Wierdl states that ergodic averages along primes converge almost everywhere for $L^{p}$-functions, $p>1$, with a polynomial version by Wierdl and Nair. Using an anti-correlation result for the von Mangoldt function due to Green and Tao, we observe everywhere convergence of such averages for nilsystems and continuous functions.

Keywords

ergodic averages along primesnilsystemseverywhere convergence

MSC classification

Primary: 37A30: Ergodic theorems, spectral theory, Markov operators 37A45: Relations with number theory and harmonic analysis
Secondary: 11B30: Arithmetic combinatorics; higher degree uniformity 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 10 , October 2020 , pp. 2769 - 2777
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2019

References

Auslander, L., Green, L. and Hahn, F.. Flows on Homogeneous Spaces (Annals of Mathematics Studies, 53) . Princeton University Press, Princeton, NJ, 1963.Google Scholar
Bergelson, V., Host, B. and Kra, B.. Multiple recurrence and nilsequences. Invent. Math. 160 (2005), 261–303.CrossRefGoogle Scholar
Bergelson, V. and Leibman, A.. Distribution of values of bounded generalized polynomials. Acta Math. 198 (2007), 155–230.Google Scholar
Bergelson, V., Leibman, A. and Lesigne, E.. Intersective polynomials and the polynomial Szemerédi theorem. Adv. Math. 219 (2008), 369–388.Google Scholar
Bergelson, V., Leibman, A. and Ziegler, T.. The shifted primes and the multidimensional Szemerédi and polynomial van der Waerden theorems. C. R. Math. Acad. Sci. Paris 349 (2011), 123–125.Google Scholar
Bourgain, J.. An approach to pointwise ergodic theorems. Geometric Aspects of Functional Analysis (1986/87) (Lecture Notes in Mathematics, 1317) . Springer, Berlin, 1988, pp. 204–223.Google Scholar
Bourgain, J.. On the pointwise ergodic theorem on L p for arithmetic sets. Israel J. Math. 61 (1988), 73–84.CrossRefGoogle Scholar
Bourgain, J.. Pointwise ergodic theorems for arithmetic sets. Publ. Math. Inst. Hautes Études Sci. 169 (1989), 5–45.Google Scholar
Chu, Q.. Convergence of weighted polynomial multiple ergodic averages. Proc. Amer. Math. Soc. 137 (2009), 1363–1369.Google Scholar
Conze, J. P. and Lesigne, E.. Sur un théorème ergodique pour des mesures diagonales. C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), 491–493.Google Scholar
Eisner, T. and Zorin-Kranich, P.. Uniformity in the Wiener–Wintner theorem for nilsequences. Discrete Contin. Dyn. Syst. 33 (2013), 3497–3516.CrossRefGoogle Scholar
Frantzikinakis, N.. Multiple correlation sequences and nilsequences. Invent. Math. 202 (2015), 875–892.Google Scholar
Frantzikinakis, N. and Host, B.. Weighted multiple ergodic averages and correlation sequences. Ergod. Th. & Dynam. Sys. 38 (2018), 81–142.CrossRefGoogle Scholar
Frantzikinakis, N., Host, B. and Kra, B.. Multiple recurrence and convergence for sequences related to the prime numbers. J. Reine Angew. Math. 611 (2007), 131–144.Google Scholar
Frantzikinakis, N., Host, B. and Kra, B.. The polynomial multidimensional Szemerédi theorem along shifted primes. Israel J. Math. 194 (2013), 331–348.Google Scholar
Furstenberg, H.. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Anal. Math. 31 (1977), 204–256.Google Scholar
Furstenberg, H. and Weiss, B.. A mean ergodic theorem for [[()[]mml:mfrac[]()]][[()[]mml:mrow []()]]1[[()[]/mml:mrow[]()]] [[()[]mml:mrow []()]]N[[()[]/mml:mrow[]()]][[()[]/mml:mfrac[]()]]∑n=1 N f (T n x)g (T n 2 x). Convergence in Ergodic Theory and Probability (Ohio State University Mathematical Research Institute Publications, 5) . De Guyter, Berlin, 1996, pp. 193–227.Google Scholar
Green, B. and Tao, T.. Linear equations in primes. Ann. of Math. (2) 171 (2010), 1753–1850.Google Scholar
Green, B. and Tao, T.. The quantitative behaviour of polynomial orbits on nilmanifolds. Ann. of Math. (2) 175 (2012), 465–540.CrossRefGoogle Scholar
Green, B. and Tao, T.. The Möbius function is strongly orthogonal to nilsequences. Ann. of Math. (2) 175 (2012), 541–566.CrossRefGoogle Scholar
Green, B., Tao, T. and Ziegler, T.. An inverse theorem for the Gowers U s+1 -norm. Ann. of Math. (2) 176 (2012), 1231–1372.CrossRefGoogle Scholar
Green, L. W.. Spectra of nilmanifolds. Bull. Amer. Math. Soc. (N.S.) 67 (1961), 414–415.Google Scholar
Host, B. and Kra, B.. An odd Furstenberg–Szemerédi theorem and affine systems. J. Anal. Math. 86 (2002), 183–220.CrossRefGoogle Scholar
Host, B. and Kra, B.. Nonconventional ergodic averages and nilmanifolds. Ann. of Math. (2) 161 (2005), 397–488.CrossRefGoogle Scholar
Host, B. and Kra, B.. Uniformity seminorms on l ∞ and applications. J. Anal. Math. 108 (2009), 219–276.CrossRefGoogle Scholar
Krause, B.. Polynomial ergodic averages converge rapidly: variations on a theorem of Bourgain. Preprint, 2014, arXiv:1402.1803.Google Scholar
Leibman, A.. Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold. Ergod. Th. & Dynam. Sys. 25 (2005), 201–213.Google Scholar
Leibman, A.. Convergence of multiple ergodic averages along polynomials of several variables. Israel J. Math. 146 (2005), 303–315.Google Scholar
Leibman, A.. Nilsequences, null-sequences, and multiple correlation sequences. Ergod. Th. & Dynam. Sys. 35 (2015), 176–191.Google Scholar
Lesigne, E.. Sur une nil-variété, les parties minimales associées à une translation sont uniquement ergodiques. Ergod. Th. & Dynam. Sys. 11 (1991), 379–391.CrossRefGoogle Scholar
Nair, R.. On polynomials in primes and Bourgain’s circle method approach to ergodic theorems. Ergod. Th. & Dynam. Sys. 11 (1991), 485–499.CrossRefGoogle Scholar
Nair, R.. On polynomials in primes and Bourgain’s circle method approach to ergodic theorems II. Studia Math. 105 (1993), 207–233.CrossRefGoogle 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.CrossRefGoogle Scholar
Szemerédi, E.. On sets of integers containing no k elements in arithmetic progression. Acta Arith. 27 (1975), 199–245 a collection of articles in memory of Juriĭ Vladimirovič Linnik.CrossRefGoogle Scholar
Thouvenot, J.-P.. La convergence presque sûre des moyennes ergodiques suivant certaines sous-suites d’entiers (d’après Jean Bourgain) [Almost sure convergence of ergodic means along some subsequences of integers (after Jean Bourgain)]. Sémin. Bourbaki, Vol. 1989/90, Astérisque No. 189–190 (1990), Exp. No. 719, 133–153 (in French).Google Scholar
Wierdl, M.. Pointwise ergodic theorem along the prime numbers. Israel J. Math. 64 (1988), 315–336.Google Scholar
Wierdl, M.. Almost everywhere convergence and recurrence along subsequences in ergodic theory. PhD Thesis, Ohio State University, 1989.Google Scholar
Wooley, T. and Ziegler, T.. Multiple recurrence and convergence along the primes. Amer. J. Math. 134 (2012), 1705–1732.CrossRefGoogle Scholar
Ziegler, T.. A non-conventional ergodic theorem for a nilsystem. Ergod. Th. & Dynam. Sys. 25 (2005), 1357–1370.Google Scholar
Ziegler, T.. Universal characteristic factors and Furstenberg averages. J. Amer. Math. Soc. 20 (2007), 53–97.Google Scholar
Zorin-Kranich, P.. Variation estimates for averages along primes and polynomials. J. Funct. Anal. 268 (2015), 210–238.CrossRefGoogle Scholar
Zorin-Kranich, P.. A double return times theorem. Preprint, 2015, arXiv:1506.05748v1.pdf.Google Scholar