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

TL;DR

This paper extends Arak's inequalities to analyze the concentration of weighted sums of i.i.d. variables, revealing new insights into the arithmetic structure of coefficients in the Littlewood-Offord problem.

Contribution

It introduces more general and precise applications of Arak's inequalities, advancing understanding of the Littlewood-Offord problem beyond previous inverse principles.

Findings

Derived new bounds for concentration functions of weighted sums.

Connected Arak's inequalities with inverse principles in additive combinatorics.

Provided refined results on the influence of coefficient structure.

Abstract

Let $X,X_1,\ldots,X_n$ 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 recent papers of Eliseeva, G\"otze 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.