Exact order of extreme L p discrepancy of infinite sequences in arbitrary dimension

Abstract

We study the extreme $L_p$ discrepancy of infinite sequences in the d-dimensional unit cube, which uses arbitrary sub-intervals of the unit cube as test sets. This is in contrast to the classical star $L_p$ discrepancy, which uses exclusively intervals that are anchored in the origin as test sets. We show that for any dimension d and any $p>1$ , the extreme $L_p$ discrepancy of every infinite sequence in ${[0,1)}^d$ is at least of order of magnitude ${(logN)}^{d/2}$ , where N is the number of considered initial terms of the sequence. For $p\in (1,\infty )$ , this order of magnitude is best possible.

Keywords: Extreme L p -discrepancy; Lower bounds; Van der Corput sequence.