Resilience for the Littlewood-Offord Problem

TL;DR

This paper investigates how many variables in a sum of random signs can be adversarially changed without causing the sum to concentrate on a specific value, extending classical Littlewood-Offord results.

Contribution

It introduces and asymptotically solves a resilience version of the Littlewood-Offord problem, analyzing adversarial influence on concentration probabilities.

Findings

Asymptotic solution to the resilience problem

Quantitative bounds on adversarial modifications

Identification of open problems in the area

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.