New applications of Arak's inequalities to the Littlewood-Offord problem

Abstract

Let $X,X_1,\dots ,X_n$ be independent identically distributed random variables. In this paper we study the behavior of concentration functions of weighted sums ${\sum }_{k=1}^n{X}_k{a}_k$ 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 context of the Littlewood-Offord problem.

23 pages. arXiv admin note: text overlap with arXiv:1506.09034

References in corpus (1)

• Bound for the maximal probability in the Littlewood-Offord problem

Cited by in corpus (1)

• A new bound in the Littlewood--Offord problem