Optimal inverse Littlewood-Offord theorems

Let ηi, i=1,…,n, be iid Bernoulli random variables, taking values ±1 with probability 12. Given a multiset V of n integers v1,…,vn, we define the concentration probability asρ(V):=supxP(v1η1+⋯+vnηn=x). A 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 and 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.