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∈ℝn satisfying |w|2=1 and |w|∞≤β with sufficiently small β>0, we establish the uniform error bound |𝔼|∑i=1nwiXi|−√(2/π)|=O(β2) 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.
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BibTeX
@misc{yu2026sharp,
title = {{A Sharp Small-Coefficient Variant of Khintchine's Inequality and the Sharp $π/2$ Theorem}},
author = {Lei Yu},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29703},
eprint = {2608.29703},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29703},
}
RIS
TY - UNPB
TI - A Sharp Small-Coefficient Variant of Khintchine's Inequality and the Sharp π/2 Theorem
AU - Lei Yu
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29703
UR - https://arxiv.org/abs/2608.29703
AB - 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∈ℝ^n satisfying |w|_2=1 and |w|_∞≤β with sufficiently small β>0, we establish the uniform error bound |𝔼|∑_i=1^nw_iX_i|−√(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.
ER -