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


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.


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

Mathematics Subject Classification

60G50 11P70 60E07 60E10 60E15 



We are grateful to anonymous reviewers for useful remarks.


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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

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