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arXiv · 2109.06461

Exact order of extreme $L_p$ discrepancy of infinite sequences in arbitrary dimension

Ralph Kritzinger·Friedrich Pillichshammer
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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 $(\log N)^{d/2}$, where $N$ is the number of considered initial terms of the sequence. For $p \in (1,\infty)$ this order of magnitude is best possible.

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Ralph Kritzinger, Friedrich Pillichshammer. 2021-09-14. Exact order of extreme $L_p$ discrepancy of infinite sequences in arbitrary dimension. https://arxiv.org/abs/2109.06461

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