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

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 unif