### 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.

Original language | English (US) |
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Pages (from-to) | 292-312 |

Number of pages | 21 |

Journal | Advances in Mathematics |

Volume | 319 |

DOIs | |

State | Published - Oct 15 2017 |

### Keywords

- Anti-concentration
- Littlewood–Offord
- Resilience

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Bandeira, A. S., Ferber, A., & Kwan, M. (2017). Resilience for the Littlewood–Offord problem.

*Advances in Mathematics*,*319*, 292-312. https://doi.org/10.1016/j.aim.2017.08.031