AI 中文总结
研究将离线Beck--Fiala猜想有效性扩展到\(d\geq\log(T)^{1 + o(1)}\)，通过结合在线Komlós问题相关思想的在线算法最小化前缀差异，解决了Spencer设定下在线向量平衡问题，且该结果本质最优。
AI 中文摘要
Beck--Fiala猜想断言，每个矩阵\(A\in\{0,1\}^{n\times T}\)，每列最多有\(d\)个非零元素，其差异为\(O(\sqrt{d})\)。Bansal和Jiang的一个重大突破结果最近证明了该猜想对于\(d\geq\log(T)^2\)成立。本文将经典的离线Beck--Fiala猜想的有效性扩展到\(d\geq\log(T)^{1 + o(1)}\)；此外，主要成果是通过一种高效的在线算法获得的，该算法可最小化前缀差异。该结果在本质上也是最优的，因为已知对于\(d = o(\log T)\)，在线前缀差异的规模为\(\omega(\sqrt{d})\)。作为直接推论，Spencer设定下的在线向量平衡的开放问题也得到了解决。该算法基于一个紧支撑的Metropolis定点游走，它结合了最近关于在线Komlós问题的几项工作的思想。证明是在与ChatGPT 5.6 Pro的对话中生成的；作者在几轮提示中提供了高层次指导，随后进行了人工检查和证明重写。
英文摘要
The Beck--Fiala conjecture asserts that every matrix $A\in\{0,1\}^{n\times T}$ with at most $d$ nonzero entries in each column has discrepancy $O(\sqrt d)$. A major breakthrough result of Bansal and Jiang recently established the validity of the conjecture for $d \ge \log(T)^2$. The present article extends the validity of the classical \textit{offline} Beck--Fiala conjecture to $d \ge \log(T)^{1+o(1)}$; moreover, the main thrust of the result is that it is actually obtained by an efficient \textit{online} algorithm that minimizes prefix discrepancy. The result is also essentially optimal, since online prefix discrepancy is known to scale as $ω(\sqrt{d})$ for $d =o(\log T)$. As an immediate corollary, the open question of online vector balancing in the Spencer setting is also resolved. The algorithm is based on a compactly supported Metropolis fixed-point walk, constructed by combining ideas from several recent works on the online Komlós problem. The proof was generated in conversation with ChatGPT 5.6 Pro; the authors provided high-level guidance in several rounds of prompting, followed by manual checking and rewriting of the proof.
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