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

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 $\sum_{k=1}^{n}X_ka_k $ with respect to the arithmetic structure of coefficients~$a_k$ in the context of the Littlewood--Offord problem. In our recent papers, 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 improved (more general and more precise) consequences of Arak's inequalities applying our new bound in the Littlewood--Offord problem. Moreover, we also obtain an improvement of the estimates used in Rudelson and Vershynin's least common denominator method.