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

2021/09/14 by Ralph Kritzinger, Kritzinger, Ralph, Friedrich Pillichshammer +1
Mathematics · #11K06 #11K31 #11K38 #FOS: Mathematics #Mathematical Approximation and Integration #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2109.06461

openalex publication_date 2021/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∈ (1,∞) this order of magnitude is best possible.

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