发表机构
Faculty of Science, Yamagata University（山形大学理学部）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对锚定盒的 bracketing 覆盖与 δ-覆盖，证明其下界并构造 bracketing 覆盖，结合下界得到 bracketing 数的渐近等价结果，还得到两类覆盖数的显式上界。
AI 中文摘要
bracketing 覆盖与 δ-覆盖为定义星差的锚定盒提供有限离散化。设 N[](d,δ) 和 N(d,δ) 分别为对应的 bracketing 数与覆盖数，我们证明下界：N[](d,δ)≥⌈δ⁻ᵈ⌉，N(d,δ)≥⌈(d!/dᵈ)δ⁻ᵈ⌉。对每个固定 d，我们构造 bracketing 覆盖，结合下界得 δ→0 时 N[](d,δ)=(1+o_d(1))δ⁻ᵈ。该构造结合粗划分与依赖盒的各向异性局部网格，其共享顶点生成渐近上系数为1的δ-覆盖，同时得到两类数的显式上界。
英文摘要
Bracketing covers and $δ$-covers provide finite discretizations of the anchored boxes that define the star discrepancy. Let $N_{[]}(d,δ)$ and $N(d,δ)$ denote the corresponding bracketing and covering numbers. We prove the lower bounds \[ N_{[]}(d,δ)\ge \lceil δ^{-d}\rceil, \qquad N(d,δ)\ge \left\lceil \frac{d!}{d^d}\,δ^{-d}\right\rceil. \] We give two explicit constructions of bracketing covers. For every fixed $d$, together with the lower bound they imply $N_{[]}(d,δ)=(1+o_d(1))δ^{-d}$ as $δ\downarrow0$. A first construction uses box-dependent anisotropic local grids and gives simple explicit bounds. A second, homothetic logarithmic-shell construction again attains this coefficient and gives $\limsup_{d\to\infty}N_{[]}(d,δ)^{1/d}\leδ^{-1}+e+O(δ)$ as $δ\downarrow0$. Combining these finite estimates with Gnewuch's general bracketing bound and a Hoeffding--Bernstein chaining argument shows that, for every $d,n\in\mathbb N$, there exists an $n$-point set with star discrepancy at most $2.3463\sqrt{d/n}$. Consequently, $\lceil5.5052d\varepsilon^{-2}\rceil$ points suffice for star discrepancy at most $\varepsilon$.
