An Upper Bound of the Minimal Dispersion via Delta Covers

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 a delta-cover with cardinality vert Gammaδ vert we prove that there is a point set, such that the largest volume of such a test set without any point is bounded by fraclog vert Gammaδ vertn + δ. For axis-parallel boxes on the unit cube this leads to a volume of at most frac4dnlog(frac9nd) and on the torus to frac4dnlog (2n).