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

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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 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.

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Acknowledgements

We are grateful to anonymous reviewers for useful remarks.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501, Bielefeld, Germany

    Friedrich Götze

  2. St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, 191023, St. Petersburg, Russia

    Andrei Yu. Zaitsev

  3. St. Petersburg State University, Universitetskaya nab. 7/9, 199034, St. Petersburg, Russia

    Andrei Yu. Zaitsev

Authors
  1. Friedrich Götze
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  2. Andrei Yu. Zaitsev
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Correspondence to Andrei Yu. Zaitsev.

Additional information

The authors were supported by the SFB 701 in Bielefeld and by SPbGU-DFG grant 6.65.37.2017. The second author was supported by grant RFBR 16-01-00367 and by the Program of the Presidium of the Russian Academy of Sciences No. 01 ‘Fundamental Mathematics and its Applications’ under grant PRAS-18-01.

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Götze, F., Zaitsev, A.Y. New applications of Arak’s inequalities to the Littlewood–Offord problem. European Journal of Mathematics 4, 639–663 (2018). https://doi.org/10.1007/s40879-018-0215-3

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  • DOI: https://doi.org/10.1007/s40879-018-0215-3

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