Expected dispersion of uniformly distributed points

TL;DR

This paper investigates the expected dispersion of random point sets in the unit cube, providing bounds and showing how the number of points needed to achieve a certain dispersion scales with dimension.

Contribution

It offers new bounds on the expected dispersion of uniformly distributed points and analyzes how the required number of points depends on dimension.

Findings

Expected dispersion bounds depend on number of points and dimension.

Minimal points needed for a given dispersion grow linearly with dimension.

Provides theoretical insights into random point set dispersion in high dimensions.

Abstract

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 $d$.