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

Abstract

Consider a random sum η1υ1...η nυn, where η1,...,η n are independently and identically distributed i.i.d. random signs and v1, ..., vn are integers. The Littlewood-Offord problem asks to maximize concentration probabilities such as P(η1υ1...η nυn = 0) subject to various hypotheses on υ1,..., υn. In this paper we develop an inverse Littlewood-Offord theory (somewhat in the spirit of Freiman's inverse theory in additive combinatorics), which starts with the hypothesis that a concentration probability is large, and concludes that almost all of the υ1,..., υn are efficiently contained in a generalized arithmetic progression. As an application we give a new bound on the magnitude of the least singular value of a random Bernoulli matrix, which in turn provides upper tail estimates on the condition number.