揭示Rademacher和的四阶矩与高阶矩的深层关系，解决多个长期未解猜想。
Fourth-Moment Geometry of Rademacher Sums
- 基于四阶质量分析，建立高阶矩的精确界限。
- 在p≥5时确定L_p/L_4 Khintchine常数的最优值，均匀向量为极值点。
- 适用于稀疏信号与随机投影，提供敏感的尾部概率估计。
设ε₁,…,εₙ为独立的Rademacher符号，a=(a₁,…,aₙ)∈ℝⁿ满足归一化条件。对归一化的Rademacher和，本文确定其高阶矩如何依赖于四阶质量。结合一个精确的固定q阶矩包络与低于凸性阈值的独立论证，得到该线性-q界在全范围p≥4下的高斯稳定性不等式。相同的四阶框架也确定了p≥5时有限维L_p/L_4 Khintchine常数的最优值，其中均匀系数向量为极值。这些结果解决了Jakimiuk以及Barański、Murawski、Nayar、Oleszkiewicz提出的猜想。此外还证明了Jakimiuk在p=3时的二次稳定性估计猜想。所得界限保留了稀疏性与有效维度的信息，适用于Rademacher随机投影和随机符号误差；这些应用在此未进一步展开。其拉普拉斯变换形式亦给出关于系数敏感的尾部界。证明过程在ChatGPT 5.6 Sol的大量协助下完成。
原文摘要 · Abstract (English)
Let $\varepsilon_1,\ldots,\varepsilon_n$ be independent Rademacher signs and let $a=(a_1,\ldots,a_n)\in\R^n$ satisfy the normalization below. For the normalized Rademacher sum, we determine how its higher moments depend on the fourth-order mass. Combining a sharp fixed-q moment envelope with a separate argument below the convexity threshold gives the Gaussian stability inequality for the full range $p\geq4$ of this linear-in-q bound. The same fourth-order framework determines the sharp finite dimensional $L_p/L_4$ Khintchine constant for $p\geq5$, with the flat coefficient vector as the extremizer. These results settle the conjectures of Jakimiuk and of Barański, Murawski, Nayar, and Oleszkiewicz stated below. We also prove Jakimiuk's conjectured quadratic stability estimate at $p=3$. The resulting bounds retain information about sparsity and effective dimension, with applications to Rademacher random projections and randomly signed errors; those applications are not developed further here. Their Laplace-transform form also gives coefficient-sensitive tail bounds. The proofs are discovered with substantial assistance from ChatGPT 5.6 Sol.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。