向量平衡：通过方向全变差
Vector Balancing via Directional Total Variation
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中文总结 AI 辅助
本文通过方向全变差技术，证明了Komlós符号问题的$3\sqrt{2\pi}$界，并推导出集合系统二染色不平衡度的平方根上界，验证了Beck-Fiala猜想。
中文摘要 AI 辅助
我们的主要结果是Komlós符号问题的一个$3\sqrt{2\pi}$界：每个由欧几里得范数至多为1的实数向量组成的有限族，都允许一个$\ell_\infty$范数小于该常数的符号和，且该界与维数和族大小无关。对于任意$\kappa\ge0$，如果一个有界开凸集在每个单位方向上支持一个方向全变差至多为$\kappa$的概率密度，那么其开集Banaszczyk变换支持另一个具有相同$\kappa$的密度，前提是平移向量$v$满足$\kappa\\|v\\|_2\le1/3$。因此，每个有限集合系统（其中每个元素至多属于$t$个集合，$t\ge1$为整数）都允许一种二染色，使得每个集合中的不平衡度小于$3\sqrt{2\pi t}$。这给出了Beck-Fiala猜想所预测的平方根依赖关系。该证明由Odin自动人工智能研究代理发现。
英文摘要
Our main result is a $3\sqrt{2π}$ bound for the Komlós signing problem: every finite family of real vectors of Euclidean norm at most one admits a signed sum of $\ell_\infty$-norm less than this constant, independently of the dimension and the family size. For any $κ\ge0$, if a bounded open convex set supports a probability density with directional total variation at most $κ$ in every unit direction, then its open-set Banaszczyk transform supports another such density with the same $κ$, provided the translation vector $v$ satisfies $κ\|v\|_2\le1/3$. As a consequence, every finite set system in which each element belongs to at most $t$ sets, where $t\ge1$ is an integer, admits a two-coloring whose imbalance in each set is less than $3\sqrt{2πt}$. This gives the square-root dependence predicted by the Beck-Fiala conjecture. The proof was discovered by the Odin Automatic AI Research Agent.
发表机构
- Harvard T.H. Chan School of Public Health（哈佛陈曾熙公共卫生学院）
机构由 AI 辅助整理，请以论文原文为准。
