Skip to main content
SpringerLink
Log in

Extension of Wiener-Wintner double recurrence theorem to polynomials

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

We extend our result on the convergence of double recurrence Wiener-Wintner averages to the case of a polynomial exponent. We show that there exists a unique set of full measure for which the averages

$$\frac{1}{N}\sum\limits_{n = 1}^N {{f_1}\left( {{T^{an}}x} \right){f_2}\left( {{T^{bn}}x} \right)\phi \left( {p\left( n \right)} \right)} $$

converge for all polynomials p with real coefficients and all complex-valued continuous functions ϕ on the unit circle T. We also show that if either function belongs to an orthogonal complement of an appropriate Host-Kra-Ziegler factor that depends on the degree of the polynomial p, the averages converge to 0 uniformly for all polynomials. This paper combines the authors’ previously announced work.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. H. Anzai, Ergodic skew product transformations on the torus, OsakaMath. J. 3 (1951), 83–99.

    MathSciNet  MATH  Google Scholar 

  2. I. Assani, Wiener Wintner Ergodic Theorems, World Science Publ. Co. Inc., River Edge, NJ, 2003.

    Book  MATH  Google Scholar 

  3. I. Assani, Pointwise double recurrence and nilsequences, Preprint. Available from arXiv:1504.05732, 2015.

  4. I. Assani, D. Duncan, and R. Moore, Pointwise characteristic factors for Wiener-Wintner double recurrence theorem Ergodic Theory Dynam. Systems 36 (2016), 1037–1066.

    MATH  Google Scholar 

  5. I. Assani and R. Moore, Extension of Wiener-Wintner double recurrence theorem to polynomials, Preliminary version, arXiv:1408.3064, 2014.

    Google Scholar 

  6. I. Assani and R. Moore, Extension of Wiener-Wintner double recurrence theorem to polynomials II, Preprint, arXiv:1409.0463, 2014.

    Google Scholar 

  7. I. Assani and R. Moore, A good universal weight for nonconventional ergodic averages in norm, Ergodic Theory Dynam. Systems 37 (2017), 1009–1025.

    Article  MathSciNet  MATH  Google Scholar 

  8. I. Assani and R. Moore, A good universal weight for multiple recurrence averages with commuting transformations in norm, Preprint, available on arXiv:1506.05370, June 2015.

    Google Scholar 

  9. V. Bergelson, B. Host, and B. Kra, Multiple recurrence and nilsequences, Invent. Math. 160 (2005), 261–303.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Bourgain, Double recurrence and almost sure convergence, J. reine angew. Math 404 (1990), 140–161.

    MathSciNet  MATH  Google Scholar 

  11. T. Eisner and B. Krause, (Uniform) convergence of twisted ergodic averages, Ergodic Theory Dynam. Systems 36 (2016), 2172–2202.

    Article  MathSciNet  MATH  Google Scholar 

  12. T. Eisner and P. Zorin-Kranich, Uniformity in the Wiener-Wintner theorem for nilsequences, Discrete and Continuous Dynamical Systems 33 (2013), 3497–3516.

    Article  MathSciNet  MATH  Google Scholar 

  13. N. Frantzikinakis, Uniformity in the polynomial Wiener-Wintner theorem, Ergodic Theory Dynam. Systems 26 (2006), 1061–1071.

    Article  MathSciNet  MATH  Google Scholar 

  14. W. T. Gowers, A new proof of Szemerédi’s theorem, Geom. Funct. Anal. 11 (2001), 465–588.

    Article  MathSciNet  MATH  Google Scholar 

  15. B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Ann. of Math. (2) 161 (2005), 387–488.

    Article  MathSciNet  MATH  Google Scholar 

  16. B. Host and B. Kra, Uniformity seminorms on ℓ ∞ and applications, J. Anal. Math. 108 (2009), 219–276.

    Article  MathSciNet  MATH  Google Scholar 

  17. L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences. John Wiley and Sons, 1974.

    MATH  Google Scholar 

  18. A. Leibman, Pointwise convergence of ergodic averages for polynomial sequence of translations on a nilmanifold, Ergodic Theory Dynam. Systems 25 (2005), 201–213.

    Article  MathSciNet  MATH  Google Scholar 

  19. E. Lesigne, Un théorème de disjonction de systèmes dynamiques et une généralisation du théorème ergodique de Wiener-Wintner, Ergodic Theory Dynam. Systems 10 (1990), 513–521.

    Article  MathSciNet  MATH  Google Scholar 

  20. E. Lesigne, Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes, Ergodic Theory Dynam. Systems 13 (1993), 767–784.

    MathSciNet  MATH  Google Scholar 

  21. T. Ziegler, Universal characteristic factors and Furstenberg averages, J. Amer. Math. Soc. 20 (2006), 53–97.

    Article  MathSciNet  MATH  Google Scholar 

  22. P. Zorin-Kranich, A nilsequence Wiener-Wintner theorem for bilinear ergodic averages. Preprint. Available from arXiv:1504.04647, 2015.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of North Carolina At Chapel Hill, Chapel Hill, NC, 27599, USA

    Idris Assani

  2. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile

    Ryo Moore

Authors
  1. Idris Assani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ryo Moore
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Idris Assani.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Assani, I., Moore, R. Extension of Wiener-Wintner double recurrence theorem to polynomials. JAMA 134, 597–613 (2018). https://doi.org/10.1007/s11854-018-0019-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-018-0019-x

Search

Navigation