{"title":"Local concentration inequalities and Tomaszewski’s conjecture","authors":"Nathan Keller, Ohad Klein","doi":"10.1145/3406325.3451011","DOIUrl":null,"url":null,"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.","PeriodicalId":132752,"journal":{"name":"Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing","volume":"7 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3406325.3451011","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
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.
我们证明了Tomaszewski猜想(1986):设f:{−1,1}n→∈的形式为f(x)=∑i=1n ai xi。则Pr[|f(x)|≤√Var[f]]≥1/2。我们的主要新工具是局部集中不等式和改进的Berry-Esseen不等式，用于离散立方体上的一次函数。这些工具是独立的兴趣，可能是有用的研究线性阈值函数和低次布尔函数。
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