关于Komlós问题的$\u007e\u007bO\u007d(\log^{1/4} n)$界的阐述
An Exposition of the $\widetilde{O}(\log^{1/4} n)$ Bound for the Komlós Problem
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中文总结 AI 辅助
本文研究Komlós猜想相关的矩阵组合偏差界问题，证明列范数不超过1的实矩阵组合偏差上界为$O((\log n)^{1/4}(\log\log n)^{7/4})$，首次改进了Banaszczyk的$O(\sqrt{\log n})$界，并反驳了Hajela的下界猜想。
中文摘要 AI 辅助
Komlós猜想指出，任何列的欧几里得范数不超过1的矩阵$A\in\mathbb R^{m\times n}$的组合偏差都受一个通用常数约束。我们证明了每一个这类矩阵的组合偏差至多为$O((\log n)^{1/4}(\log\log n)^{7/4})$。这是对Banaszczyk[Banaszczyk, Random Struct. Algorithms, 1998]建立的$O(\sqrt{\log n})$界的首次渐近改进，同时也反驳了Hajela[Hajela, European J. Combin., 1988]提出的存在$Ω(\sqrt{\log n})$阶下界的猜想。
英文摘要
A conjecture of Komlós states that the combinatorial discrepancy of any matrix $A\in\mathbb R^{m\times n}$ whose columns have Euclidean norm at most one is bounded by a universal constant. We prove that the combinatorial discrepancy of every such matrix is at most $O((\log n)^{1/4}(\log\log n)^{7/4})$. This is the first asymptotic improvement over the $O(\sqrt{\log n})$ bound established by Banaszczyk [Banaszczyk, Random Struct.\ Algorithms, 1998], and it refutes a conjecture of Hajela [Hajela, European J.\ Combin., 1988] that a lower bound of order $Ω(\sqrt{\log n})$ should hold.
