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

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 Pn={x1,…,xn}⊂[0,1]d and k∈{0,1,…,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.