Arak Inequalities for Concentration Functions and the Littlewood--Offord Problem

Littlewood-Offord Inequalities for Random Variables

The concentration of a real-valued random variable X is c(X) sup P(t < X < + 1). Given bounds on the concentrations of n independent random variables, how large can the concentration of their sum be? The main aim of this paper is to give a best possible upper bound for the concentration of the sum of n independent random variables, each of concentration at most 1/k, where k is an integer. Other...

Resilience for the Littlewood-Offord Problem

In this paper we study a resilience version of the classical Littlewood-Offord problem. Consider the sum X(ξ) = ∑n i=1 aiξi, where a = (ai) n i=1 is a sequence of non-zero reals and ξ = (ξi) n i=1 is a sequence of i.i.d. random variables with Pr[ξi = 1] = Pr[ξi = −1] = 1/2. Motivated by some problems from random matrices, we consider the following question for any given x: how many of the ξi is...

Erratum for “inverse Littlewood-offord Problem“

Bilinear and Quadratic Variants on the Littlewood-offord Problem

If f(x1, . . . , xn) is a polynomial dependent on a large number of independent Bernoulli random variables, what can be said about the maximum concentration of f on any single value? For linear polynomials, this reduces to one version of the classical Littlewood-Offord problem: Given nonzero constants a1, . . . , an, what is the maximum number of sums of the form ±a1±a2± · · · ± an which take o...

The Littlewood-offord Problem and Invertibility of Random Matrices

We prove two basic conjectures on the distribution of the smallest singular value of random n×n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n−1/2, which is optimal for Gaussian matrices. Moreover, we give a optimal estimate on the tail probability. This comes as a consequence of a new and essentially sharp estimate in...