A sharp inverse Littlewood-Offord theorem

TL;DR

This paper characterizes multisets with high concentration probability for sums of Bernoulli variables, providing an optimal inverse Littlewood-Offord theorem that unifies and extends previous bounds.

Contribution

It offers an asymptotically optimal inverse theorem for the concentration probability of multisets, improving understanding of when large concentration occurs.

Findings

Characterization of multisets with large concentration probability

Strengthening and unifying previous Littlewood-Offord results

Optimal bounds for sum concentration of Bernoulli variables

Abstract

Let $\eta_i, i=1,..., n$ be iid Bernoulli random variables. Given a multiset $\bv$ of $n$ numbers $v_1, ..., v_n$, the \emph{concentration probability} $\P_1(\bv)$ of $\bv$ is defined as $\P_1(\bv) := \sup_{x} \P(v_1 \eta_1+ ... v_n \eta_n=x)$. A classical result of Littlewood-Offord and Erd\H os from the 1940s asserts that if the $v_i $ are non-zero, then this probability is at most $O(n^{-1/2})$. Since then, many researchers obtained better bounds by assuming various restrictions on $\bv$. In this paper, we give an asymptotically optimal characterization for all multisets $\bv$ having large concentration probability. This allow us to strengthen or recover several previous results in a straightforward manner.