Point sets with optimal order of extreme and periodic discrepancy

TL;DR

This paper investigates the extreme and periodic Lp discrepancy of point sets in high-dimensional cubes, establishing the optimal order of their minimal discrepancy as a function of the number of points.

Contribution

It extends previous work by analyzing the general Lp discrepancy case and determines the exact order of minimal discrepancy in terms of the number of points for fixed dimensions.

Findings

Minimal discrepancy order is (log N)^{(d-1)/2} for fixed dimension d.

Provides exact formulas for discrepancy in the L2 case and extends to general Lp.

Clarifies relations between different discrepancy measures in high dimensions.

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 $(\log N)^{(d-1)/2}$.