欣钦不等式的小系数变体与精确π/2定理
A Sharp Small-Coefficient Variant of Khintchine's Inequality and the Sharp $π/2$ Theorem
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中文总结 AI 辅助
本文针对有界最大系数的归一化加权拉德马赫和，证明了精确二次正态近似的误差界，结合零偏斯坦方法与小球概率估计验证了收敛速率最优性，并将其应用于改进布尔函数的FKN定理（精确π/2定理）。
中文摘要 AI 辅助
我们针对具有有界最大系数的归一化加权拉德马赫和的一阶绝对矩，证明了一种改进的二次正态近似。对任意满足‖w‖₂=1且‖w‖_∞≤β（β>0足够小）的权重向量w∈ℝⁿ，我们在所有容许权重构型上建立了一致误差界|E[|∑ᵢ₌₁ⁿwᵢXᵢ|]−√(2/π)|=O(β²)。我们的证明结合零偏斯坦方法与改进的小球概率估计，以利用对称性抵消并控制绝对值检验函数的非光滑残差。一个显式的极值构造进一步验证了该二次收敛速率的最优性。作为应用，我们为布尔函数的弗里德古特-卡莱-内奥（FKN）定理（亦称为精确π/2定理）建立了渐近精确的改进，刻画了远离独裁者的函数的1阶傅里叶能量。
英文摘要
We prove a refined quadratic normal approximation for the first absolute moment of normalized weighted Rademacher sums with bounded maximal coefficients. For any weight vector $w\in\mathbb{R}^{n}$ satisfying $\|w\|_{2}=1$ and $\|w\|_{\infty}\leqβ$ with sufficiently small $β>0$, we establish the uniform error bound $|\mathbb{E}|\sum_{i=1}^{n}w_{i}X_{i}|-\sqrt{2/π}|=O(β^{2})$ over all admissible weight configurations. Our proof combines zero-bias Stein's method and refined small-ball probability estimates to exploit symmetry cancellation and control the non-smooth residual of the absolute-value test function. An explicit extremal construction further verifies the optimality of this quadratic convergence rate. As an application, we establish an asymptotically sharp refinement of the Friedgut--Kalai--Naor (FKN) theorem for Boolean functions, also known as the sharp $π/2$ theorem, characterizing the level-1 Fourier energy for functions deviating far from dictatorships.
发表机构
- Nankai University（南开大学）
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