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

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

3 Scopus citations

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

Bibliographical note

Publisher Copyright:
© 2022 Elsevier Inc.

Funding

Research supported by the Israel Science Foundation (grants no. 1612/17 and 2669/21) and by the U.S-Israel Binational Science Foundation (grant no. 2014290).

Funders	Funder number
United States-Israel Binational Science Foundation	2014290
Israel Science Foundation	2669/21, 1612/17

Keywords

Analysis of Boolean functions
Combinatorics
Probabilistic inequalities
Tail inequalities

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10.1016/j.aim.2022.108558

Cite this

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KW - Combinatorics

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KW - Tail inequalities

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Proof of Tomaszewski's conjecture on randomly signed sums

Abstract

Bibliographical note

Funding

Keywords

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