Given a finite set $A \subseteq \mathbb{R}^d$, points $a_1,a_2,\dotsc,a_{\ell} \in A$ form an $\ell$-hole in A if they are the vertices of a convex polytope, which contains no points of A in its interior. We construct arbitrarily large point sets in general position in $\mathbb{R}^d$ having no holes of size $O(4^dd\log d)$ or more. This improves the previously known upper bound of order $d^{d+o(d)}$ due to Valtr. The basic version of our construction uses a certain type of equidistributed point sets, originating from numerical analysis, known as (t,m,s)-nets or (t,s)-sequences, yielding a bound of $2^{7d}$. The better bound is obtained using a variant of (t,m,s)-nets, obeying a relaxed equidistribution condition.

中文翻译：

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关于 d 维点集中的凸孔

给定一个有限集$A \subseteq \mathbb{R}^d$, 点$a_1,a_2,\dotsc,a_{\ell} \in A$形成一个$\ell$- 打洞一种如果它们是凸多面体的顶点，其中不包含一种在它的内部。我们在一般位置构造任意大的点集$\mathbb{R}^d$没有大小孔$O(4^dd\log d)$或者更多。这改进了先前已知的阶数上限$d^{d+o(d)}$由于 Valtr。我们构造的基本版本使用某种类型的等分布点集，源自数值分析，称为 (吨,米,s)-网或 (吨,s)-序列，产生一个界限$2^{7d}$. 使用 (吨,米,s)-nets，服从一个宽松的等分布条件。