Inverse Littlewood-Offord Theorems and the Condition Number of Random Discrete Matrices

Consider a random sum $\eta _1 v_1 \, + \,.\,.\,.\, + \,\eta _n v_{n,} $ where $\eta _1 ,\,.\,.\,.,\,\eta _n$ are independently and identically distributed (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 $v_1 ,\,.\,.\,.,\,v_n $ . In this paper we develop an inverse Littlewood-Oiford 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 $v_1 ,\,.\,.\,.,\,v_n $ 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.