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An ergodic correspondence principle, invariant means and applications

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Abstract

A theorem due to Hindman states that if E is a subset of ℕ with d*(E) > 0, where d* denotes the upper Banach density, then for any ε > 0 there exists N ∈ ℕ such that \(d^{\ast}(\cup_{i=1}^{N}(E-i))>1-\varepsilon\). Curiously, this result does not hold if one replaces the upper Banach density d* with the upper density \(\bar{d}\). Originally proved combinatorially, Hindman’s theorem allows for a quick and easy proof using an ergodic version of Furstenberg’s correspondence principle. In this paper, we establish a variant of the ergodic Furstenberg’s correspondence principle for general amenable (semi)-groups and obtain some new applications, which include a refinement and a generalization of Hindman’s theorem and a characterization of countable amenable minimally almost periodic groups.

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References

  1. M. Beiglböck, V. Bergelson and A. Fish, Sumset phenomenon in countable amenable groups, Advances in Mathematics 223 (2010), 416–432.

    Article  MathSciNet  Google Scholar 

  2. V. Bergelson, Ergodic Ramsey theory, in Logic and Combinatorics, Contemporary Mathematics, Vol. 65, American Mathematical Society, Providence, RI, 1987, pp. 63–87.

    Chapter  Google Scholar 

  3. V. Bergelson, Ergodic theory and Diophantine problems, in Topics in Symbolic Dynamics and Applications, London Mathematical Society Lecture Note Series, Vol. 279, Cambridge University Press, Cambridge, 2000, pp. 167–205.

    MATH  Google Scholar 

  4. V. Bergelson, J. C. Christopherson, D. Robertson and P. Zorin-Kranich, Finite products sets and minimally almost periodic groups, Journal of Functional Analysis 270 (2016), 2126–2167.

    Article  MathSciNet  Google Scholar 

  5. V. Bergelson and H. Furstenberg, WM groups and Ramsey theory, Topology and its Applications 156 (2009), 2572–2580.

    Article  MathSciNet  Google Scholar 

  6. V. Bergelson and D. Glasscock, On the interplay between additive and multiplicative largeness and its combinatorial applications, Journal of Combinatorial Theory. Series A 172 (2020), Article no. 105203.

  7. V. Bergelson, B. Host and B. Kra, Multiple recurrence and nilsequences, Inventiones Mathematicaer 160 (2005), 261–303.

    Article  MathSciNet  Google Scholar 

  8. V. Bergelson and I. J. Håland Knutson, Weak mixing implies weak mixing of higher orders along tempered functions, Ergodic Theory and Dynamical Systems 29 (2009), 1375–1416.

    Article  MathSciNet  Google Scholar 

  9. V. Bergelson, G. Kolesnik, M. Madritsch, Y. Son and R. Tichy, Uniform distribution of prime powers and sets of recurrence and van der Corput sets in ℤk, Israel Journal of Mathematics 201 (2014), 729–760.

    Article  MathSciNet  Google Scholar 

  10. V. Bergelson, G. Kolesnik and Y. Son, Uniform distribution of subpolynomial functions along primes and applications, Journal d’Analyse Mathématique 137 (2019), 135–187.

    Article  MathSciNet  Google Scholar 

  11. V. Bergelson and A. Leibman, Cubic averages and large intersections, in Recent Trends in Ergodic Theory and Dynamical Systems, Contemporary Mathematics, Vol. 631, American Mathematical Society, Providence, RI, 2015, pp. 5–19.

    MATH  Google Scholar 

  12. V. Bergelson and E. Lesigne, Van der Corput sets in ℤd, Colloquium Mathematicum 110 (2008), 1–49.

    Article  MathSciNet  Google Scholar 

  13. V. Bergelson and R. McCutcheon, Recurrence for semigroup actions and a non-commutative Schur theorem, in Topological Dynamics and Applications, Contemporary Mathematics, Vol. 215, American Mathematical Society, Providence, RI, 1998, pp. 205–222.

    Chapter  Google Scholar 

  14. G. Birkhoff, A note on topological groups, Compositio Mathematica 3 (1936), 427–430.

    MathSciNet  MATH  Google Scholar 

  15. M. Boshernitzan, G. Kolesnik, A. Quas and M. Wierdl, Ergodic averaging sequences, Journal d’Analyse Mathématique 95 (2005), 63–103.

    Article  MathSciNet  Google Scholar 

  16. R. B. Burckel, Weakly Almost Periodic Functions on Semigroups, Gordon and Breach Science Publishers, New York-London-Paris, 1970.

    MATH  Google Scholar 

  17. T. Downarowicz, D. Huczek and G. Zhang, Tilings of amenable groups, Jourmal für die Reine und Angewandte Mathematik 747 (2019), 277–298.

    Article  MathSciNet  Google Scholar 

  18. W. F. Eberlein, Abstract ergodic theorems and weak almost periodic functions, Transactions of the American Mathematical Society 67 (1949), 217–240.

    Article  MathSciNet  Google Scholar 

  19. N. Frantzikinakis, Ergodicity of the Liouville system implies the Chowla conjecture, Discrete Analysis 2017 (2017), Article no. 19.

  20. H. Furstenberg, Prediction Theory, Ph.D. Thesis, Princeton University, Princeton, NJ, 1958.

    MATH  Google Scholar 

  21. H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Mathematical Systems Theory 1 (1967), 1–49.

    Article  MathSciNet  Google Scholar 

  22. H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, Journal d’Analyse Mathématique 31 (1977), 204–256.

    Article  MathSciNet  Google Scholar 

  23. H. Furstenberg, Poincaré recurrence and number theory, Bulletin of the American Mathematical Society 5 (1981) 211–234.

    Article  Google Scholar 

  24. H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lecture, Princeton University Press, Princeton, NJ, 1981.

    Book  Google Scholar 

  25. H. Furstenberg, Y. Katznelson and D. Ornstein, The ergodic theoretical proof of Szemerédi’s theorem, Bulletin of the American Mathematical Society 7 (1982) 527–552.

    Article  Google Scholar 

  26. F. P. Greenleaf, Invariant Means on Topological Groups and Their Applications, Van Nostrand Mathematical Studies, Vol. 16, Van Nostrand Reinhold, New York-Toronto, ON-London, 1969.

    MATH  Google Scholar 

  27. L. Gillman and M. Jerison, Rings of Continuous Functions, The University Series in Higher Mathematics, D. Van Nostrand, Princeton, NJ-Toronto, ON-London-New York, 1960.

    Book  Google Scholar 

  28. A. Gomilko, M. Lemanczyk and T. de la Rue, On Furstenberg systems of aperiodic multiplicative functions of Matomaki, Radziwill and Tao, https://arxiv.org/abs/2006.09958.

  29. N. Hindman, Finite sums from sequences within cells of a partition of N, Journal of Combinatorial Theory. Series A 17 (1974) 1–11.

    Article  MathSciNet  Google Scholar 

  30. N. Hindman, On density, translates, and pairwise sums of integers, Journal of Combinatorial Theory. Series A 33 (1982) 147–157.

    Article  MathSciNet  Google Scholar 

  31. N. Frantzikinakis and B. Host, The logarithmic Sarnak conjecture for ergodic weights, Annals of Mathematics 187 (2018) 869–931.

    Article  MathSciNet  Google Scholar 

  32. N. Frantzikinakis and B. Host, Furstenberg systems of bounded multiplicative functions and applications, International Mathematics Research Notices 2021 (2021) 6077–6107.

    Article  MathSciNet  Google Scholar 

  33. R. I. Jewett, The prevalence of uniquely ergodic systems, Journal of Mathematics and Mechanics 19 (1969/1970) 717–729.

    MathSciNet  MATH  Google Scholar 

  34. S. Kakutani, Selected Papers. Vol. 2, Contemporary Mathematicians, Birkhäuser, Boston, MA, 1986.

    MATH  Google Scholar 

  35. W. Krieger, On unique ergodicity, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II: Probability Theory, University of California Press, Berkeley, CA, 1972 pp. 327–346.

    Google Scholar 

  36. I. Namioka, Følner’s conditions for amenable semi-groups, Mathematica Scandinavica 15 (1964) 18–28.

    Article  MathSciNet  Google Scholar 

  37. A. L. T. Paterson, Amenability, Mathematical Surveys and Monographs, Vol. 29, American Mathematical Society, Providence, RI, 1988.

    Book  Google Scholar 

  38. A. Rosenthal, Strictly ergodic models and amenable group action, Ph.D. Thesis, Ubiversité Paris 6, Paris, 1986.

    Google Scholar 

  39. K. Schmidt, Asymptotic properties of unitary representations and mixing, Proceedings of the London Mathematical Society 48 (1984) 445–460.

    Article  MathSciNet  Google Scholar 

  40. E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arithmetica 27 (1975) 199–245.

    Article  MathSciNet  Google Scholar 

  41. V. S. Varadarajan, Groups of automorphisms of Borel spaces, Transactions of the American Mathematical Society 109 (1963) 191–220.

    Article  MathSciNet  Google Scholar 

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Acknowledgment

The authors would like to thank the referee to whom the present version of this paper owes a great deal.

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  1. Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA

    Vitaly Bergelson & Andreu Ferré Moragues

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  1. Vitaly Bergelson
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  2. Andreu Ferré Moragues
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Correspondence to Andreu Ferré Moragues.

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Bergelson, V., Moragues, A.F. An ergodic correspondence principle, invariant means and applications. Isr. J. Math. 245, 921–962 (2021). https://doi.org/10.1007/s11856-021-2233-y

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  • DOI: https://doi.org/10.1007/s11856-021-2233-y

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