Let ηi, i = 1, . . . , n be iid Bernoulli random variables, taking values ±1 with probability 1 2 . Given a multiset V of n integers v1, . . . , vn, we define the concentration probability asA classical result of Littlewood-Offord and Erdős from the 1940s asserts that, if the vi are non-zero, then ρ(V ) is O(n −1/2 ). Since then, many researchers have obtained improved bounds by assuming various extra restrictions on V .About 5 years ago, motivated by problems concerning random matrices, Tao and Vu introduced the Inverse Littlewood-Offord problem. In the inverse problem, one would like to characterize the set V , given that ρ(V ) is relatively large.In this paper, we introduce a new method to attack the inverse problem. As an application, we strengthen the previous result of Tao and Vu, obtaining an optimal characterization for V . This immediately implies several classical theorems, such as those of Sárközy-Szemerédi and Halász.The method also applies to the continuous setting and leads to a simple proof for the β-net theorem of Tao and Vu, which plays a key role in their recent studies of random matrices.All results extend to the general case when V is a subset of an abelian torsion-free group, and ηi are independent variables satisfying some weak conditions. n i=1 v i η i . The concentration probability is defined to be ρ(V ) := sup x P(S = x).Motivated by their study of random polynomials in the 1940s, Littlewood and Offord [7] raised the question of bounding ρ(V ). (We call this the forward Littlewood-Offord problem, in contrast with the inverse Littlewood-Offord problem discussed in the next section.) They 2000 Mathematics Subject Classification. 11B25.

Abstract. Let ξ be a real random variable with mean zero and variance one and A = {a1, . . . , an} be a multi-set in R d . The random sum SA := a1ξ1 + · · · + anξn where ξi are iid copies of ξ is of fundamental importance in probability and its applications.We discuss the small ball problem, the aim of which is to estimate the maximum probability that SA belongs to a ball with given small radius, following the discovery made by Littlewood-Offord and Erdős almost 70 years ago. We will mainly focus on recent developments that characterize the structure of those sets A where the small ball probability is relatively large. Applications of these results include full solutions or significant progresses of many open problems in different areas.

Gaps (or spacings) between consecutive eigenvalues are a central topic in random matrix theory. The goal of this paper is to study the tail distribution of these gaps in various random matrix models. We give the first repulsion bound for random matrices with discrete entries and the first super-polynomial bound on the probability that a random graph has simple spectrum, along with several applications. Mathematics Subject Classification

Let $A_n$ be an $n$ by $n$ random matrix whose entries are independent real random variables with mean zero, variance one and with subexponential tail. We show that the logarithm of $|\det A_n|$ satisfies a central limit theorem. More precisely, \begin{eqnarray*}\sup_{x\in {\mathbf {R}}}\biggl|{\mathbf {P}}\biggl(\frac{\log(|\det A_n|)-({1}/{2})\log (n-1)!}{\sqrt{({1}/{2})\log n}}\le x\biggr)-{\mathbf {P}}\bigl(\mathbf {N}(0,1)\le x\bigr)\biggr|\\\qquad\le\log^{-{1}/{3}+o(1)}n.\end{eqnarray*}Comment: Published in at http://dx.doi.org/10.1214/12-AOP791 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Let Mn denote a random symmetric n by n matrix, whose upper diagonal entries are iid Bernoulli random variables (which take value −1 and 1 with probability 1/2). Improving the earlier result by Costello, Tao and Vu [4], we show that Mn is nonsingular with probability 1 − O(n −C ) for any positive constant C. The proof uses an inverse Littlewood-Offord result for quadratic forms, which is of interest of its own.