A Sharp Small-Coefficient Variant of Khintchine's Inequality and the Sharp $\pi /2$ Theorem

Abstract

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 $\Vert w{\Vert }_2=1$ and $\Vert w{\Vert }_{\infty }\le \beta$ with sufficiently small $\beta >0$, we establish the uniform error bound $|\mathbb{E}|{\sum }_{i=1}^n{w}_i{X}_i|-\sqrt{2/\pi }|=O({\beta }^2)$ 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 $\pi /2$ theorem, characterizing the level-1 Fourier energy for functions deviating far from dictatorships.

arXiv comment10 pages