发表机构
The University of Hong Kong（香港大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文研究拉德马赫和的四阶矩几何，结合矩包络与凸性阈值论证得到高斯稳定性不等式，解决相关猜想并证明p=3时的二次稳定性估计，所得边界可应用于拉德马赫随机投影等领域。
AI 中文摘要
设ε₁,…,εₙ为独立拉德马赫符号，a=(a₁,…,aₙ)∈ℝⁿ满足下述归一化条件。针对归一化拉德马赫和，我们确定其高阶矩如何依赖于四阶质量。将尖锐的固定q矩包络与凸性阈值以下的单独论证相结合，得到了对线性于q的边界的全范围p≥4成立的高斯稳定性不等式。相同的四阶框架确定了p≥5时尖锐的有限维Lₚ/L₄欣钦常数，其中平坦系数向量为极值元。这些结果解决了Jakimiuk以及Barański、Murawski、Nayar和Oleszkiewicz提出的下述猜想。我们还证明了Jakimiuk在p=3时猜想的二次稳定性估计。所得边界保留了稀疏性和有效维数的信息，可应用于拉德马赫随机投影和随机符号误差；这些应用在此处不再进一步展开。其拉普拉斯变换形式也给出了系数敏感的尾边界。证明过程在ChatGPT 5.6 Sol的大量协助下完成。
英文摘要
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.
