发表机构
Gaoling School of Artificial Intelligence, Renmin University of China（中国人民大学高瓴人工智能学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了平面中点集相对于轴平行矩形的组合差异的最坏情况为 $\Theta(\log^{3/2}n)$，填补了已知下界与上界之间的差距，并给出了简单初等的证明。
AI 中文摘要
我们证明了平面中 $n$ 个点相对于轴平行矩形的组合差异的最坏情况为 $\Theta(\log^{3/2}n)$。已知的界限为 $\Omega(\log n)$ 和 $O(\log^{3/2}n)$；我们证明了匹配的下界。该下界对随机点集成立：对于每个 $A>0$，存在常数 $c_A>0$，使得以至少 $1-e^{-An}$ 的概率，单位正方形中 $n$ 个独立均匀点的每个着色都有一个锚定矩形，其不平衡度至少为 $c_A(\log_2n)^{3/2}$。证明出人意料地简单且初等。它逐位揭示一个坐标数字。在点上具有压倒性概率的情况下，一个振荡势的条件增益在 $\Theta(\log n)$ 个数字上累加为 $\Omega(\log^{3/2}n)$。有界差异控制波动足够好，以便对所有着色进行联合界。
英文摘要
We show that the worst-case combinatorial discrepancy of $n$ points in the plane with respect to axis-parallel rectangles is $Θ(\log^{3/2}n)$. The known bounds were $Ω(\log n)$ and $O(\log^{3/2}n)$; we prove the matching lower bound. It holds for random point sets: for every $A>0$, there is a constant $c_A>0$ such that, with probability at least $1-e^{-An}$, every coloring of $n$ independent uniform points in the unit square has an anchored rectangle with imbalance at least $c_A(\log_2n)^{3/2}$. The proof is surprisingly simple and elementary. It reveals one coordinate digit by digit. With overwhelming probability over the points, the conditional gains of an oscillation potential add up to $Ω(\log^{3/2}n)$ over $Θ(\log n)$ digits. Bounded differences control the fluctuations well enough for a union bound over all colorings.
Comments18 pages, 3 figures
