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arXiv · 1807.01492

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

Aicke Hinrichs·Joscha Prochno·Mario Ullrich·Jan Vybiral
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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.

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Aicke Hinrichs, Joscha Prochno, Mario Ullrich, Jan Vybiral. 2018-07-04. The minimal $k$-dispersion of point sets in high-dimensions. https://doi.org/10.1016/j.jco.2018.10.001

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