Local Concentration Inequalities and Tomaszewski’s Conjecture

Nathan Keller; Ohad Klein

doi:10.1145/3406325.3451011

Local Concentration Inequalities and Tomaszewski’s Conjecture

Nathan Keller
, Ohad Klein

Department of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

1 Scopus citations

Abstract

We prove Tomaszewski's conjecture (1986): Let f:{-1,1}n ? R be of the form f(x)= ?i=1n ai xi. Then Pr[|f(x)| ? Var[f]] ? 1/2. Our main novel tools are local concentration inequalities and an improved Berry-Esseen inequality for first-degree functions on the discrete cube. These tools are of independent interest, and may be useful in the study of linear threshold functions and of low degree Boolean functions.

Original language	English
Title of host publication	STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing
Editors	Samir Khuller, Virginia Vassilevska Williams
Publisher	Association for Computing Machinery
Pages	1656-1669
Number of pages	14
ISBN (Electronic)	9781450380539
DOIs	https://doi.org/10.1145/3406325.3451011
State	Published - 15 Jun 2021
Event	53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021 - Virtual, Online, Italy Duration: 21 Jun 2021 → 25 Jun 2021

Publication series

Name	Proceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)	0737-8017

Conference

Conference	53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021
Country/Territory	Italy
City	Virtual, Online
Period	21/06/21 → 25/06/21

Bibliographical note

Publisher Copyright:
© 2021 ACM.

Funding

The Research was supported by the Israel Science Foundation (grant no. 1612/17) and by the Binational US-Israel Science Foundation (grant no. 2014290).

Funders	Funder number
Israel Science Foundation	2014290, 1612/17

Keywords

Combinatorics
Probabilistic Inequalities
Tail Inequalities

Access to Document

10.1145/3406325.3451011

Cite this

Keller, N., & Klein, O. (2021). Local Concentration Inequalities and Tomaszewski’s Conjecture. In S. Khuller, & V. V. Williams (Eds.), STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (pp. 1656-1669). (Proceedings of the Annual ACM Symposium on Theory of Computing). Association for Computing Machinery. https://doi.org/10.1145/3406325.3451011

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abstract = "We prove Tomaszewski's conjecture (1986): Let f:\{-1,1\}n ? R be of the form f(x)= ?i=1n ai xi. Then Pr[|f(x)| ? Var[f]] ? 1/2. Our main novel tools are local concentration inequalities and an improved Berry-Esseen inequality for first-degree functions on the discrete cube. These tools are of independent interest, and may be useful in the study of linear threshold functions and of low degree Boolean functions.",

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Keller, N & Klein, O 2021, Local Concentration Inequalities and Tomaszewski’s Conjecture. in S Khuller & VV Williams (eds), STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing. Proceedings of the Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, pp. 1656-1669, 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, Virtual, Online, Italy, 21/06/21. https://doi.org/10.1145/3406325.3451011

Local Concentration Inequalities and Tomaszewski’s Conjecture. / Keller, Nathan; Klein, Ohad.
STOC 2021 - Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing. ed. / Samir Khuller; Virginia Vassilevska Williams. Association for Computing Machinery, 2021. p. 1656-1669 (Proceedings of the Annual ACM Symposium on Theory of Computing).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Local Concentration Inequalities and Tomaszewski’s Conjecture

Abstract

Publication series

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Bibliographical note

Funding

Keywords

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