A sharp inverse Littlewood-Offord theorem

Abstract

Let $η_i, i=1,..., n$ be iid Bernoulli random variables. Given a multiset $\bv$ of $n$ numbers $v_1, ..., v_n$, the \emph{concentration probability} $¶_1(\bv)$ of $\bv$ is defined as $¶_1(\bv) := \sup_{x} ¶(v_1 η_1+ ... v_n η_n=x)$. 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.

17 pages, no figures, to appear, Random Structures and Algorithms. This is the final version, incorporating the referee's corrections and suggestions