Proof of Tomaszewski's conjecture on randomly signed sums

Nathan Keller; Ohad Klein

doi:10.1016/j.aim.2022.108558

Proof of Tomaszewski's conjecture on randomly signed sums

Nathan Keller, Ohad Klein

Department of Mathematics

Bar-Ilan University

Research output: Contribution to journal › Article › peer-review

Abstract

We prove the following conjecture, due to Tomaszewski (1986): Let X=∑_i=1ⁿa_ix_i, where ∑_ia_i²=1 and each x_i 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.

Original language	English
Article number	108558
Journal	Advances in Mathematics
Volume	407
DOIs	https://doi.org/10.1016/j.aim.2022.108558
State	Published - 8 Oct 2022

Keywords

Analysis of Boolean functions
Combinatorics
Probabilistic inequalities
Tail inequalities

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1016/j.aim.2022.108558

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title = "Proof of Tomaszewski's conjecture on randomly signed sums",

abstract = "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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AB - 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.

KW - Analysis of Boolean functions

KW - Combinatorics

KW - Probabilistic inequalities

KW - Tail inequalities

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DO - 10.1016/j.aim.2022.108558

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VL - 407

JO - Advances in Mathematics

JF - Advances in Mathematics

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Proof of Tomaszewski's conjecture on randomly signed sums

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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