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 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 a lower bound of the expected dispersion. In particular, we show that the minimal number of points required to achieve an expected dispersion less than $\epsilon \in (0,1)$ depends linearly on the dimension $d$.

中文翻译：

---

均匀分布点的预期分散

设置点的离散度 ${[0，\mathrm{1个}]}^d$是单位立方体内不与点集相交的最大轴平行框的体积。我们研究关于随机集合的预期色散$ñ$由均匀分布的随机变量的iid序列确定的点。取决于点数$ñ$ 和尺寸 $d$我们提供了预期色散的上限和下限。特别是，我们显示出达到预期色散所需的最小点数小于$\epsilon \in （0，\mathrm{1个}）$ 线性取决于尺寸 $d$。