The minimal $k$-dispersion of point sets in high-dimensions

TL;DR

This paper extends the concept of minimal dispersion to include the volume of the largest box with at most k points in high-dimensional unit cubes, providing bounds that match classical dispersion results for many k values.

Contribution

It introduces the concept of k-dispersion, extending classical dispersion, and derives bounds that align with known results across a wide range of k.

Findings

Established upper and lower bounds for minimal k-dispersion.

Bounds coincide with classical dispersion bounds for many k values.

Provides insights into high-dimensional point set distributions.

Abstract

In this manuscript we introduce and study an extended version of the minimal dispersion of point sets, which has recently attracted considerable attention. Given a set $\mathscr P_n=\{x_1,\dots,x_n\}\subset [0,1]^d$ and $k\in\{0,1,\dots,n\}$, we define the $k$-dispersion to be the volume of the largest box amidst a point set containing at most $k$ points. The minimal $k$-dispersion is then given by the infimum over all possible point sets of cardinality $n$. We provide both upper and lower bounds for the minimal $k$-dispersion that coincide with the known bounds for the classical minimal dispersion for a surprisingly large range of $k$'s.