An Upper Bound of the Minimal Dispersion via Delta Covers

Abstract

For a point set of n elements in the d-dimensional unit cube and a class of test sets we are interested in the largest volume of a test set which does not contain any point. For all natural numbers n, d and under the assumption of the existence of a δ-cover with cardinality |Γδ| we prove that there is a point set, such that the largest volume of such a test set without any point is bounded above by \(\frac {\log \vert \varGamma _\delta \vert }{n} + \delta \). For axis-parallel boxes on the unit cube this leads to a volume of at most \(\frac {4d}{n}\log (\frac {9n}{d})\) and on the torus to \(\frac {4d}{n}\log (2n)\).