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

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 ${\mathcal{P}}_n=\{x_1,\dots ,x_n\}\subset [0,1{]}^d$ 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.

9 pages

References in corpus (4)

• A lower bound for the dispersion on the torus
• An upper bound on the minimal dispersion
• A note on the dispersion of admissible lattices
• Dispersion of the Fibonacci and the Frolov point sets