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arXiv · 1609.08136

Resilience for the Littlewood-Offord Problem

Afonso S. Bandeira·Asaf Ferber·Matthew Kwan
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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.

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Afonso S. Bandeira, Asaf Ferber, Matthew Kwan. 2016-09-26. Resilience for the Littlewood-Offord Problem. https://arxiv.org/abs/1609.08136

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