Inverse Littlewood-Offord theorems and the condition number of random discrete matrices

Abstract

Consider a random sum ${\eta }_1{v}_1+...+{\eta }_n{v}_n$, where ${\eta }_1,...,{\eta }_n$ are i.i.d. random signs and $v_1,...,v_n$ are integers. The Littlewood-Offord problem asks to maximize concentration probabilities such as $\P ({\eta }_1{v}_1+...+{\eta }_n{v}_n=0)$ subject to various hypotheses on the $v_1,...,v_n$. In this paper we develop an \emph{inverse} Littlewood-Offord theorem (somewhat in the spirit of Freiman's inverse sumset theorem), which starts with the hypothesis that a concentration probability is large, and concludes that almost all of the $v_1,...,v_n$ are efficiently contained in an arithmetic progression. As an application we give some new bounds on the distribution of the least singular value of a random Bernoulli matrix, which in turn gives upper tail estimates on the condition number.

37 pages, no figures, to appear, Annals of Math. Referee comments incorporated