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 a $delta$-cover with cardinality $|{\Gamma }_{\delta }|$ we prove that there is a point set, such that the largest volume of such a test set without any point is bounded by $\frac{\log |{\Gamma }_{\delta }|}{n}+\delta$. 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 $\frac{4d}{n}\log (2n)$.

arXiv comment10 pages, accepted in Ian Sloan's 80th birthday festschrift