A Sharp Small-Coefficient Variant of Khintchine's Inequality and the Sharp $π/2$ 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 $\|w\|_{2}=1$ 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.