发表机构
Technical University of Munich（慕尼黑工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出布尔矩阵级数的新小球不等式，结合高斯互反估计、方向变差符号定理和复制论证证明，并由此给出 Kadison-Singer 问题的新证明及多个偏差理论问题的快速推论。
AI 中文摘要
我们证明了布尔矩阵级数的新小球不等式。主要例子是 $\mathbb E_s[{\text{det}(I-S^2)^\beta\\,\mathbf 1_{\{\\|S\\|<1\}}}]\ge e^{-O(\beta\tau)}$，该不等式适用于由对称矩阵 $A_1,\dots,A_n$ 和均匀随机符号 $s\in\{\pm1\}^n$ 构成的布尔矩阵级数 $S=\sum_i s_iA_i$。具体而言，当 $\beta\ge1$ 且 $\tau=\sum_i\text{Tr} A_i^2$ 时，只要 $(\text{Tr} A_i^2)_{i=1}^n$ 的最大值和某个方差项被通用常数上界所限制，该不等式即成立。证明结合了 (Akbas 和 Sra 2026) 的高斯互反估计、(Guo, Fang, 和 Lu 2026) 的方向变差符号定理，以及一个将存在性转化为良好符号上的 Gibbs 定律的复制论证。最值得注意的是，布尔小球为 Kadison-Singer 问题（最一般情形）提供了新的、无交错性质的证明；它还能快速推出 Matrix Spencer 和 Komlós 定理作为推论，同时为多种偏差理论问题提供了超过六个几乎直接的证明。
英文摘要
We prove new small-ball inequalities for boolean matrix-series. The leading example is $\mathbb E_s[{\text{det}(I-S^2)^β\,\mathbf 1_{\{\|S\|<1\}}}]\ge e^{-O(βτ)}$, which holds for boolean matrix-series $S=\sum_i s_iA_i$ formed using symmetric matrices $A_1,\dots,A_n$ and uniformly random signs $s\in\{\pm1\}^n$. Specifically, this inequality holds for all $β\ge1$ with $τ=\sum_i\text{Tr} A_i^2$, as soon as the maximum of $(\text{Tr} A_i^2)_{i=1}^n$ and a certain variance term are bounded above by universal constants. The proof combines the Gaussian reciprocal estimate of (Akbas and Sra 2026), the directional-variation signing theorem of (Guo, Fang, and Lu 2026), and a replica argument that turns existence into a Gibbs law on good signings. Most notably, boolean small-ball delivers a new, interlacing-free proof of Kadison-Singer (most general case); it also recovers Matrix Spencer and Komlós as quick corollaries, while yielding more than six almost immediate proofs of an assortment of discrepancy theoretic problems.
Comments37 pages, comments welcome
