Optimal Inverse Littlewood-Offord theorems

TL;DR

This paper introduces a new method for the inverse Littlewood-Offord problem, providing an optimal characterization of sets with high concentration probability, with applications to classical theorems and random matrix theory.

Contribution

The paper presents a novel approach to the inverse Littlewood-Offord problem, achieving an optimal characterization of sets with large concentration probability and extending results to broader algebraic settings.

Findings

Established an optimal characterization for sets with high concentration probability.

Extended classical theorems like Sarkozy-Szemeredi and Halasz.

Provided a simple proof for the beta-net theorem in random matrix theory.

Abstract

Let eta_i be iid Bernoulli random variables, taking values -1,1 with probability 1/2. Given a multiset V of n integers v_1,..., v_n, we define the concentration probability as rho(V) := sup_{x} Pr(v_1 eta_1+...+ v_n eta_n=x). A classical result of Littlewood-Offord and Erdos from the 1940s asserts that if the v_i are non-zero, then rho(V) is O(n^{-1/2}). Since then, many researchers 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 give a characterization of the set V, given that rho(V) is relatively large. In this paper, we introduce a new method to attack the inverse problem. As an application, we strengthen a previous result of Tao and Vu, obtaining an optimal characterization for V. This immediately implies several classical theorems, such as those of Sarkozy-Szemeredi and Halasz. The method also applies in the continuous setting and leads to a simple proof for the beta-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 eta_i are independent variables satisfying some weak conditions.