Resilience for the Littlewood–Offord problem

Afonso S. Bandeira, Asaf Ferber, Matthew Kwan

Research output: Contribution to journalArticle

Abstract

Consider the sum X(ξ)=∑i=1 naiξi, where a=(ai)i=1 n is a sequence of non-zero reals and ξ=(ξi)i=1 n is a sequence of i.i.d. Rademacher random variables (that is, Pr⁡[ξi=1]=Pr⁡[ξi=−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 ξi 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.

Original languageEnglish (US)
Pages (from-to)292-312
Number of pages21
JournalAdvances in Mathematics
Volume319
DOIs
StatePublished - Oct 15 2017

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Keywords

  • Anti-concentration
  • Littlewood–Offord
  • Resilience

ASJC Scopus subject areas

  • Mathematics(all)

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