Advertisement

European Journal of Mathematics

, Volume 4, Issue 2, pp 639–663 | Cite as

New applications of Arak’s inequalities to the Littlewood–Offord problem

  • Friedrich Götze
  • Andrei Yu. ZaitsevEmail author
Research Article

Abstract

Let \(X_1,\ldots ,X_n\) be independent identically distributed random variables. In this paper we study the behavior of concentration functions of weighted sums Open image in new window with respect to the arithmetic structure of coefficients \(a_k\) in the context of the Littlewood–Offord problem. In recent papers of Eliseeva, Götze and Zaitsev, we discussed the relations between the inverse principles stated by Nguyen, Tao and Vu and similar principles formulated by Arak in his papers from the 1980’s. In this paper, we will derive some more general and more precise consequences of Arak’s inequalities providing new results in the Littlewood–Offord problem.

Keywords

Concentration functions Inequalities Littlewood–Offord problem Sums of independent random variables 

Mathematics Subject Classification

60G50 11P70 60E07 60E10 60E15 

Notes

Acknowledgements

We are grateful to anonymous reviewers for useful remarks.

References

  1. 1.
    Arak, T.V.: On the approximation of \(n\)-fold convolutions of distributions, having a non-negative characteristic functions with accompanying laws. Theory Probab. Appl. 25(2), 221–243 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arak, T.V.: On the convergence rate in Kolmogorov’s uniform limit theorem I. Theory Probab. Appl. 26(2), 219–239 (1982)CrossRefzbMATHGoogle Scholar
  3. 3.
    Arak, T.V., Zaitsev, A.Yu.: Uniform limit theorems for sums of independent random variables. Proc. Steklov Inst. Math. 174, 1–222 (1988)Google Scholar
  4. 4.
    Eliseeva, Yu.S., Zaitsev, A.Yu.: On the Littlewood–Offord problem. J. Math. Sci. (N.Y.) 214(4), 467–473 (2016)Google Scholar
  5. 5.
    Erdös, P.: On a lemma of Littlewood and Offord. Bull. Amer. Math. Soc. 51(12), 898–902 (1945)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Esseen, C.G.: On the Kolmogorov–Rogozin inequality for the concentration function. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 5(3), 210–216 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Freiman, G.A.: Foundations of a Structural Theory of Set Addition. Translations of Mathematical Monographs, vol. 37. American Mathematical Society, Providence (1973)Google Scholar
  8. 8.
    Götze, F., Eliseeva, Yu.S., Zaitsev, A.Yu.: Arak’s inequalities for concentration functions and the Littlewood–Offord problem. Dokl. Math. 93(2), 202–206 (2016)Google Scholar
  9. 9.
    Götze, F., Eliseeva, Yu.S., Zaitsev, A.Yu.: Arak inequalities for concentration functions and the Littlewood–Offord problem. Teor. Veroyatn. Primen. 62(2), 241–266 (2017) (in Russian)Google Scholar
  10. 10.
    Green, B.: Notes on progressions and convex geometry (2005). http://people.maths.ox.ac.uk/greenbj/papers/convexnotes.pdf
  11. 11.
    Hengartner, W., Theodorescu, R.: Concentration Functions. Probability and Mathematical Statistics. Academic Press, New York (1974)Google Scholar
  12. 12.
    Kolmogorov, A.N.: Two uniform limit theorems for sums of independent random variables. Theory Probab. Appl. 1(4), 384–394 (1956)CrossRefGoogle Scholar
  13. 13.
    Littlewood, J.E., Offord, A.C.: On the number of real roots of a random algebraic equation (III). Rec. Math. [Mat. Sbornik] N.S. 12(54)(3), 277–286 (1943)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Nguyen, H., Vu, V.: Optimal inverse Littlewood–Offord theorems. Adv. Math. 226(6), 5298–5319 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Nguyen, H.H., Vu, V.H.: Small ball probability, inverse theorems and applications. In: Lovász, L., Ruzsa, I.Z., Sós, V.T. (eds.) Erdős Centennial Proceeding, Bolyai Society Mathematical Studies, vol. 25, pp. 409–463. Bolyai Mathematical Society, Budapest (2013)Google Scholar
  16. 16.
    Petrov, V.V.: Sums of Independent Random Variables. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 82. Springer, New York (1975)Google Scholar
  17. 17.
    Tao, T., Vu, V.: Additive Combinatorics. Cambridge Studies in Advanced Mathematics, vol. 105. Cambridge University Press, Cambridge (2006)Google Scholar
  18. 18.
    Tao, T., Vu, V.: John-type theorems for generalized arithmetic progressions and iterated sumsets. Adv. Math. 219(2), 428–449 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tao, T., Vu, V.: Inverse Littlewood–Offord theorems and the condition number of random discrete matrices. Ann. Math. 169(2), 595–632 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Tao, T., Vu, V.: A sharp inverse Littlewood–Offord theorem. Random Structures Algorithms 37(4), 525–539 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zaitsev, A.Yu.: A bound for the maximal probability in the Littlewood–Offord problem. J. Math. Sci. (N.Y.) 219(5), 743–746 (2016)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  2. 2.St. Petersburg Department of Steklov Mathematical InstituteSt. PetersburgRussia
  3. 3.St. Petersburg State UniversitySt. PetersburgRussia

Personalised recommendations