On the Littlewood--Offord problem

TL;DR

This paper explores the Littlewood--Offord problem's connection to concentration functions of symmetric infinitely divisible distributions, providing multivariate generalizations and linking sum concentration to the arithmetic structure of distributions.

Contribution

It introduces multivariate generalizations of Arak's results, connecting sum concentration with the arithmetic structure of distributions' supports.

Findings

Established multivariate extensions of concentration bounds

Linked sum concentration to support arithmetic structure

Extended results to symmetric infinitely divisible distributions

Abstract

The paper deals with studying a connection of the Littlewood--Offord problem with estimating the concentration functions of some symmetric infinitely divisible distributions. Some multivariate generalizations of results of Arak (1980) are given. They show a connection of the concentration function of the sum with the arithmetic structure of supports of distributions of independent random vectors for arbitrary distributions of summands.