Point sets with optimal order of extreme and periodic discrepancy
Volume 204 / 2022
Abstract
We study the extreme and the periodic $L_p$ discrepancy of point sets in the $d$-dimensional unit cube. The extreme discrepancy uses arbitrary subintervals of the unit cube as test sets, whereas the periodic discrepancy is based on periodic intervals modulo 1. This is in contrast to the classical star discrepancy, which uses as test sets exclusively intervals that are anchored at the origin. In a recent paper the authors together with Aicke Hinrichs studied relations between the $L_2$ versions of these notions of discrepancy and presented exact formulas for typical two-dimensional quasi-Monte Carlo point sets. In this paper we study the general $L_p$ case and deduce the exact order of magnitude of the relevant minimal discrepancy in terms of the number $N$ of elements of the point sets considered, for arbitrary but fixed dimension $d$, which is $(\log N)^{(d-1)/2}$.
