### 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) |
---|---|

Pages (from-to) | 292-312 |

Number of pages | 21 |

Journal | Advances in Mathematics |

Volume | 319 |

DOIs | |

State | Published - Oct 15 2017 |

### Fingerprint

### Keywords

- Anti-concentration
- Littlewood–Offord
- Resilience

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

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

**Resilience for the Littlewood–Offord problem.** / Bandeira, Afonso; Ferber, Asaf; Kwan, Matthew.

Research output: Contribution to journal › Article

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

}

TY - JOUR

T1 - Resilience for the Littlewood–Offord problem

AU - Bandeira, Afonso

AU - Ferber, Asaf

AU - Kwan, Matthew

PY - 2017/10/15

Y1 - 2017/10/15

N2 - 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.

AB - 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.

KW - Anti-concentration

KW - Littlewood–Offord

KW - Resilience

UR - http://www.scopus.com/inward/record.url?scp=85028512055&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85028512055&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2017.08.031

DO - 10.1016/j.aim.2017.08.031

M3 - Article

AN - SCOPUS:85028512055

VL - 319

SP - 292

EP - 312

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -