The minimal $k$-dispersion of point sets in high-dimensions
This extends the concept of minimal dispersion, which is important for numerical integration and discrepancy theory, but the results are largely incremental.
The paper introduces and analyzes the minimal $k$-dispersion of point sets in high dimensions, providing upper and lower bounds that match classical dispersion bounds for a wide range of $k$.
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.