Resilience for the Littlewood–Offord problem

Consider the sum X(ξ)=∑_i=1 ⁿa_iξ_i, where a=(a_i)_i=1 ⁿ is a sequence of non-zero reals and ξ=(ξ_i)_i=1 ⁿ is a sequence of i.i.d. Rademacher random variables (that is, Pr⁡[ξ_i=1]=Pr⁡[ξ_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 ξ_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.