Functional Analysis and Its Applications

, Volume 49, Issue 2, pp 110–121 | Cite as

A quantitative version of the Beurling-Helson theorem

  • S. V. Konyagin
  • I. D. Shkredov


It is proved that any continuous function φ on the unit circle such that the sequence \({\left\{ {{e^{in\varphi }}} \right\}_{n \in \mathbb{Z}}}\) has small Wiener norm \(\left\{ {{e^{in\varphi }}} \right\} = o\left( {{{\log }^{1/22}}|n|{{\left( {\log \log |n|} \right)}^{ - 3/11}}} \right)\), \(|n| \to \infty \) is linear. Moreover, lower bounds for the Wiener norms of the characteristic functions of subsets of ℤ p in the case of prime p are obtained.


Wiener norm Beurling-Helson theorem dissociated sets 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Beurling and H. Helson, “Fourier-Stieltjes transforms with bounded powers,” Math. Scand., 1 (1953), 120–126.zbMATHMathSciNetGoogle Scholar
  2. [2]
    B. Green and S. Konyagin, “On the Littlewood problem modulo a prime,” Canad. J. Math., 61:1 (2009), 141–164.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    A. Zygmund, Trigonometric Series, vol. 2, Cambridge University Press, Cambridge, 1959.zbMATHGoogle Scholar
  4. [4]
    J.-P. Kahane, “Transformées de Fourier des fonctions sommables,” in: Proc. Internat. Congr. Math., 1962, Inst. Mittag-Leffler, Djursholm, 1963, 114–131.Google Scholar
  5. [5]
    J.-P. Kahane, Séries de Fourier absolument convergenntes, Springer-Verlag, Berlin, 1970.Google Scholar
  6. [6]
    S. V. Konyagin, “On a problem of Littlewood,” Izv. Akad. Nauk SSSR, Ser. Mat., 45:2 (1981), 243–265.zbMATHMathSciNetGoogle Scholar
  7. [7]
    V. V. Lebedev, “Quantitative estimates in Beurling-Helson type theorems,” Mat. Sb., 201:12 (2010), 103–130; English transl.: Russian Acad. Sci. Sb. Math., 201:12 (2010), 1811–1836.CrossRefGoogle Scholar
  8. [8]
    V. V. Lebedev, “Estimates in Beurling-Helson type theorems: multidimensional case,” Mat. Zametki, 90:3 (2011), 394–407; English transl.: Math. Notes, 90:3–4 (2011), 373–384.CrossRefGoogle Scholar
  9. [9]
    V. V. Lebedev, “Absolutely convergent Fourier series. An improvement of the Beurling-Helson theorem,” Funkts. Anal. Prilozhen., 46:2 (2012), 52–65; English transl.: Functional Anal. Appl., 46:2 (2012), 121–132.CrossRefGoogle Scholar
  10. [10]
    O. C. McGehee, L. Pigno, and B. Smith, “Hardy’s inequality and the L1 norm of exponential sums,” Ann. of Math., 113:3 (1981), 613–618.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    W. Rudin, Fourier Analysis on Groups, John Wiley & Sons, New York, 1990 (reprint of the 1962 original).zbMATHCrossRefGoogle Scholar
  12. [12]
    T. Sanders, “The Littlewood-Gowers problem,” J. Anal. Math., 101 (2007), 123–162.zbMATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    I. D. Shkredov, “On sets of large exponential sums,” Izv. Ross. Akad. Nauk, Ser. Mat., 72:1 (2008), 161–182; English transl.: Russian Acad. Sci. Izv. Math., 72:1 (2008), 149–168.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscow State UniversityMoscowRussia
  2. 2.Steklov Mathematical InstituteInstitute for Information Transmission of RASMoscowRussia

Personalised recommendations