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引用次数: 1
摘要
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。
Local concentration inequalities and Tomaszewski’s conjecture
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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