Israel Journal of Mathematics

, Volume 214, Issue 1, pp 109–120 | Cite as

Polynomial recurrence with large intersection over countable fields

  • Vitaly Bergelson
  • Donald Robertson


We give a short proof of polynomial recurrence with large intersection for additive actions of finite-dimensional vector spaces over countable fields on probability spaces, improving upon the known size and structure of the set of strong recurrence times.


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  1. [BD08]
    V. Bergelson and T. Downarowicz, Large sets of integers and hierarchy of mixing properties of measure preserving systems, Colloquium Mathematicum 110 (2008), 117–150.MathSciNetCrossRefMATHGoogle Scholar
  2. [Ber03]
    V. Bergelson, Minimal idempotents and ergodic Ramsey theory, in Topics in Dynamics and Ergodic Theory, London Mathematical Society Lecture Note Series, Vol. 310, Cambridge University Press, Cambridge, 2003, pp. 8–39.MathSciNetMATHGoogle Scholar
  3. [Ber10]
    V. Bergelson, Ultrafilters, IP sets, dynamics, and combinatorial number theory, in Ultrafilters Across Mathematics, Contemporary Mathematics, Vol. 530, American Mathematical Society, Providevce, RI, 2010, pp. 23–47.MathSciNetCrossRefMATHGoogle Scholar
  4. [BH90]
    V. Bergelson and N. Hindman, Nonmetrizable topological dynamics and Ramsey theory, Transactions of the American Mathematical Society 320 (1990), 293–320.MathSciNetCrossRefMATHGoogle Scholar
  5. [BH94]
    V. Bergelson and N. Hindman, On IP * sets and central sets, Combinatorica 14 (1994), 269–277.MathSciNetCrossRefMATHGoogle Scholar
  6. [BLM05]
    V. Bergelson, A. Leibman and R. McCutcheon, Polynomial Szemerédi theorems for countable modules over integral domains and finite fields, Journal d’Analyse Mathématique 95 (2005), 243–296.MathSciNetCrossRefMATHGoogle Scholar
  7. [BR15]
    V. Bergelson and D. Robertson, Polynomial multiple recurrence over rings of integers, Ergodic Theory and Dynamical Systems, 2015, Scholar
  8. [FK85]
    H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for IP-systems and combinatorial theory, Journal d’Analyse Mathématique 45 (1985), 117–168.MathSciNetCrossRefMATHGoogle Scholar
  9. [FW78]
    H. Furstenberg and B. Weiss, Topological dynamics and combinatorial number theory, Journal d’Analyse Mathématique 34 (1978), 61–85.MathSciNetCrossRefMATHGoogle Scholar
  10. [Hin74]
    N. Hindman, Finite sums from sequences within cells of a partition of N, Journal of Combinatorial Theory. Series A 17 (1974), 1–11.MathSciNetCrossRefMATHGoogle Scholar
  11. [HJ63]
    A. W. Hales and R. I. Jewett, Regularity and positional games, Transactions of the American Mathematical Society 106 (1963), 222–229.MathSciNetCrossRefMATHGoogle Scholar
  12. [HS12]
    N. Hindman and D. Strauss, Algebra in the Stone–Cech Compactification, Walter de Gruyter, Berlin, 2012.MATHGoogle Scholar
  13. [Lar98]
    P. G. Larick, Results in polynomial recurrence for actions of fields, PhD thesis, Ohio State University, Colombus, Ohio, 1998.Google Scholar
  14. [McC14]
    R. McCutcheon, Private communication, 2014.Google Scholar
  15. [MW14]
    R. McCutcheon and A. Windsor, D sets and a Sárközy theorem for countable fields, Israel Journal of Mathematics 201 (2014), 123–146.MathSciNetGoogle Scholar
  16. [MZ14]
    R. McCutcheon and J. Zhou, D sets and IP rich sets in Z, Fundamenta Mathematicae, 233 (2016), 71–82.MathSciNetGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2016

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA

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