Resilience for the Littlewood–Offord problem

Afonso 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

Fingerprint

Resilience
Open Problems
Random variable
Upper bound

Keywords

  • Anti-concentration
  • Littlewood–Offord
  • Resilience

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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

In: Advances in Mathematics, Vol. 319, 15.10.2017, p. 292-312.

Research output: Contribution to journalArticle

Bandeira, Afonso ; Ferber, Asaf ; Kwan, Matthew. / Resilience for the Littlewood–Offord problem. In: Advances in Mathematics. 2017 ; Vol. 319. pp. 292-312.
@article{53910a2fe6b14c4e92421f975be1b89c,
title = "Resilience for the Littlewood–Offord problem",
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.",
keywords = "Anti-concentration, Littlewood–Offord, Resilience",
author = "Afonso Bandeira and Asaf Ferber and Matthew Kwan",
year = "2017",
month = "10",
day = "15",
doi = "10.1016/j.aim.2017.08.031",
language = "English (US)",
volume = "319",
pages = "292--312",
journal = "Advances in Mathematics",
issn = "0001-8708",
publisher = "Academic Press Inc.",

}

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 -