Estimates for the concentration functions in the Littlewood--Offord problem

TL;DR

This paper investigates the concentration functions of weighted sums of i.i.d. random variables, emphasizing the influence of the coefficients' arithmetic structure, with implications for understanding the singular values of random matrices.

Contribution

It refines existing results on concentration functions related to the Littlewood--Offord problem, providing new bounds and insights relevant to random matrix theory.

Findings

Refined bounds on concentration functions for weighted sums

Enhanced understanding of the role of coefficient structure

Implications for singular value analysis of random matrices

Abstract

Let $X,X_1,...,X_n$ be independent identically distributed random variables. In this paper we study the behavior of the concentration functions of the weighted sums $\sum\limits_{k=1}^{n}a_k X_k$ with respect to the arithmetic structure of coefficients $a_k$. Such concentration results recently became important in connection with investigations about singular values of random matrices. In this paper we formulate and prove some refinements of a result of Vershynin (R. Vershynin, Invertibility of symmetric random matrices, arXiv:1102.0300. (2011). Published in Random Structures and Algorithms, v. 44, no. 2, 135--182 (2014)).