A sharp inverse Littlewood-Offord theorem

Let ${\eta }_i,i=1,\mathrm{.}\mathrm{.}\mathrm{.},n$ be iid Bernoulli random variables. Given a multiset $\text{bv}$ of $n$ numbers $v_1,\mathrm{.}\mathrm{.}\mathrm{.},v_n$, the \emph{concentration probability} ${\mathrm{\P }}_1(\text{bv})$ of $\text{bv}$ 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 $O(n^{-1\mathrm{/}2})$. Since then, many researchers obtained better bounds by assuming various restrictions on $\text{bv}$. In this paper, we give an asymptotically optimal characterization for all multisets $\text{bv}$ having large concentration probability. This allow us to strengthen or recover several previous results in a straightforward manner.