Point sets with optimal order of extreme and periodic discrepancy

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 sub-intervals of the unit cube as test sets, whereas the periodic discrepancy is based on periodic intervals modulo one. This is in contrast to the classical star discrepancy, which uses exclusively intervals that are anchored in the origin as test sets. 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 respective minimal discrepancy in the number $N$ of elements of the considered point sets, for arbitrary but fixed dimension $d$, which is $(\log N)^{(d-1)/2}$.