Abstract
Consider the sum X(ξ)=∑i=1naiξi, where a=(ai)i=1n is a sequence of non-zero reals and ξ=(ξi)i=1n 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]. 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, demonstrating an interesting connection to the notion of an additive basis from additive combinatorics. We also present several interesting open problems.
Original language | English (US) |
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Pages (from-to) | 93-99 |
Number of pages | 7 |
Journal | Electronic Notes in Discrete Mathematics |
Volume | 61 |
DOIs | |
State | Published - Aug 2017 |
Keywords
- Additive basis
- Anti-concentration
- Littlewood-Offord
- Resilience
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Applied Mathematics