发表机构
TU Munich, Dept. of Mathematics, School of CIT（慕尼黑工业大学，数学系，信息与计算学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文阐述Komlós猜想的新证明，利用二次狄利克雷能量得到偏差<3π，并借助对数凹性改进至C₀≈7.515，同时优化了算法版本。
AI 中文摘要
我们简要阐述了Guo、Fang和Lu（2026a）关于Komlós猜想的最新证明，其中我们提出了一种基于简单“二次狄利克雷能量”的证明，该证明给出了（较宽松的）偏差<3π。我们在附录中指出，利用对数凹性可得到（更紧的）界C₀≈7.515（小于Guo、Fang和Lu（2026a）的3√(2π)）。（我们还大幅改进了算法Komlós（Guo、Fang和Lu（2026b）），并在附录中报告。）
英文摘要
Guo, Fang, and Lu recently proved the Komlós conjecture: for vectors $v_1,\ldots,v_n\in\mathbb R^d$ of Euclidean norm at most one, there are signs $\varepsilon_j\in\{-1,1\}$ with $\|\sum_j\varepsilon_jv_j\|_\infty\le3\sqrt{2π}$. We give a short proof of the bound $3π$ that keeps the geometric lifting framework of (Guo, Fang, and Lu 2026a) while replacing the analytic core by a quadratic Dirichlet energy. Moreover, using a more fine-grained analysis of the stability of the product-cosine function of (Smirnov and Vershynin 2026) under translations, we further sharpen the bound to below $6.9013$. On the algorithmic side, based on the polynomial-time construction of (Guo, Fang, and Lu 2026b) we give a deterministic algorithm that finds a coloring of discrepancy at most $37.54$ using at most $\widetilde O(mn+n^4)$ arithmetic operations.
CommentsComments and feedback welcome. GitHub repo: https://github.com/suvrit/komlos
