The dispersion of a point set in $[0,1]^d$ is the volume of the largest axis parallel box inside the unit cube that does not intersect with the point set. We study the expected dispersion with respect to a random set of $n$ points determined by an i.i.d. sequence of uniformly distributed random variables. Depending on the number of points $n$ and the dimension $d$ we provide an upper and lower bound of the expected dispersion. In particular, we show that the minimal number of points required to achieve an expected dispersion less than $\varepsilon\in(0,1)$ depends linearly on the dimension $d$.

中文翻译：

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均匀分布点的预期离散度

$[0,1]^d$中点集的离散度是单位立方体内不与点集相交的最大轴平行盒的体积。我们研究了关于由均匀分布的随机变量的 iid 序列确定的一组随机 $n$ 点的预期分散。根据点数 $n$ 和维度 $d$，我们提供了预期离差的上限和下限。特别是，我们表明实现小于 $\varepsilon\in(0,1)$ 的预期分散所需的最小点数与维度 $d$ 线性相关。