Resilience for the Littlewood-Offord Problem

Abstract

Consider the sum $X(\xi)=\sum_{i=1}^n a_i\xi_i$, where $a=(a_i)_{i=1}^n$ is a sequence of non-zero reals and $\xi=(\xi_i)_{i=1}^n$ is a sequence of i.i.d. Rademacher random variables (that is, $\Pr[\xi_i=1]=\Pr[\xi_i=-1]=1/2$). The classical Littlewood-Offord problem asks for the best possible upper bound on the concentration probabilities $\Pr[X=x]$. In this paper we study a resilience version of the Littlewood-Offord problem: how many of the $\xi_i$ is an adversary typically allowed to change without being able to force concentration on a particular value? We solve this problem asymptotically, and present a few interesting open problems.