Resilience for the Littlewood–Offord problem

Abstract Consider the sum X ( ξ ) = ∑ i = 1 n a i ξ 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.

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