Arak Inequalities for Concentration Functions and the Littlewood--Offord Problem: a shortened version

TL;DR

This paper explores how the arithmetic structure of coefficients influences the concentration of weighted sums of i.i.d. random variables, connecting classical and modern inverse principles in the Littlewood--Offord problem.

Contribution

It relates Tao and Vu's Inverse Principle to Nguyen and Vu's and Arak's earlier work, providing a unified perspective on concentration phenomena.

Findings

Connections between different inverse principles established.

Analysis of concentration functions related to the structure of coefficients.

Discussion of implications for random matrix singular values.

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 $\sum_{k=1}^{n} X_k a_k$ with respect to the arithmetic structure of coefficients~$a_k$ in the context of the Littlewood--Offord problem. Concentration results of this type received renewed interest in connection with distributions of singular values of random matrices. Recently, Tao and Vu proposed an Inverse Principle in the Littlewood--Offord problem. We discuss the relations between the Inverse Principle of Tao and Vu as well as that of Nguyen and Vu and a similar principle formulated for sums of arbitrary independent random variables in the work of Arak from the 1980's. This paper is a shortened and edited version of the preprint arXiv:1506.09034. Here we present the results without proofs.