We have decided to discontinue the publication of preprints on our preprint server end of 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
In the first part of this paper we derive lower bounds and constructive upper bounds for the bracketing numbers of anchored and unanchored axis-parallel boxes in the
In the second part we apply these results to geometric discrepancy. We derive upper bounds for the inverse of the star and the extreme discrepancy with explicitly given small constants and an optimal dependence on the dimension
We also discuss an algorithm from [E. Thiémard, An algorithm to compute bounds for the star discrepancy, J. Complexity 17 (2001), 850-880] to approximate the star-discrepancy of a given