Advertisement

Journal of Mathematical Sciences

, Volume 218, Issue 5, pp 599–608 | Cite as

On the Wiener Norm of Subsets of ℤ p of Medium Size

  • S. V. Konyagin
  • I. D. Shkredov
Article
  • 21 Downloads

Abstract

We give a lower bound for the Wiener norm of the characteristic function of a subset A from ℤ p , where p is a prime number, in the case where exp((log p/log log p)1/3) ≤ |A| ≤ p/3.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Bourgain and M. Z. Garaev, “On a variant of sum-product estimates and explicit exponential sum bounds in prime fields,” Math. Proc. Cambridge Philos. Soc., 146, No. 1, 1–21 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    B. J. Green and S. V. Konyagin, “On the Littlewood problem modulo a prime,” Can. J. Math., 61, No. 1, 141–164 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    S. V. Konyagin, “On a problem of Littlewood,” Izv. Ross. Akad. Nauk, 45, No. 2, 243–265 (1981).MathSciNetzbMATHGoogle Scholar
  4. 4.
    S. V. Konyagin and I. D. Shkredov, Quantitative Version of the Beurling–Helson Theorem, arXiv: 1401.4429[math.CA].Google Scholar
  5. 5.
    V. V. Lebedev, “Absolutely convergent Fourier series. An improvement of the Beurling–Helson theorem,” Funkts. Anal. Prilozh., 46, No. 2, 52–65 (2012).CrossRefzbMATHGoogle Scholar
  6. 6.
    O. C. McGehee, L. Pigno, and B. Smith, “Hardy’s inequality and the L 1 norm of exponential sums,” Ann. Math., 113, 613–618 (1981).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    T. Sanders, “The Littlewood–Gowers problem,” J. Anal. Math., 101, 123–162 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    T. Sanders, “The structure theory of set addition revisited,” Bull. Am. Math. Soc., 50, No. 1, 93–127 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    T. Schoen, “New bounds in Balog–Szemerédi–Gowers theorem,” Combinatorica, accepted.Google Scholar
  10. 10.
    T. Tao and V. Vu, Additive Combinatorics, Cambridge Univ. Press, Cambridge (2006).CrossRefzbMATHGoogle Scholar
  11. 11.
    A. Zygmund, Trigonometric Series, Vol. 2, Cambridge Univ. Press, Cambridge (2002).Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia
  2. 2.Moscow State UniversityMoscowRussia
  3. 3.IITP RASMoscowRussia

Personalised recommendations