Proof of Tomaszewski's conjecture on randomly signed sums

We prove the following conjecture, due to Tomaszewski (1986): Let X=∑i=1naixi, where ∑iai2=1 and each xi is a uniformly random sign. Then Pr⁡[|X|≤1]≥1/2. Our main novel tools are local concentration inequalities and an improved Berry-Esseen inequality for Rademacher sums.

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Bibliographic Details
Published in:	Advances in mathematics (New York. 1965) Vol. 407; p. 108558
Main Authors:	Keller, Nathan, Klein, Ohad
Format:	Journal Article
Language:	English
Published:	Elsevier Inc 08.10.2022
Subjects:	Analysis of Boolean functions Combinatorics Probabilistic inequalities Tail inequalities Probabilistic inequalities Analysis of Boolean functions Tail inequalities Combinatorics
ISSN:	0001-8708, 1090-2082
Online Access:	Get full text
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Description
Summary:	We prove the following conjecture, due to Tomaszewski (1986): Let X=∑i=1naixi, where ∑iai2=1 and each xi is a uniformly random sign. Then Pr⁡[\|X\|≤1]≥1/2. Our main novel tools are local concentration inequalities and an improved Berry-Esseen inequality for Rademacher sums.
ISSN:	0001-8708 1090-2082
DOI:	10.1016/j.aim.2022.108558