A sharp inverse Littlewood-Offord theorem



Let ηi,i = 1,…,n be iid Bernoulli random variables. Given a multiset vof n numbers v1,…,vn, the concentration probability P1(v) of v is defined as P1(v) := supxP(v1η1+ …vnηn = x). A classical result of Littlewood–Offord and Erdős from the 1940s asserts that if the vi are nonzero, then this probability is at most O(n-1/2). Since then, many researchers obtained better bounds by assuming various restrictions on v.

In this article, we give an asymptotically optimal characterization for all multisets v having large concentration probability. This allow us to strengthen or recover several previous results in a straightforward manner. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010