A sharp inverse Littlewood-Offord theorem

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.