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

TL;DR

This paper establishes the exact order of the extreme $L_p$ discrepancy for infinite sequences in any dimension, showing it grows like $( ext{log} N)^{d/2}$, which is optimal for $p > 1$.

Contribution

It provides the first precise order of the extreme $L_p$ discrepancy for sequences in arbitrary dimensions, extending classical discrepancy results.

Findings

Extreme $L_p$ discrepancy grows as $( ext{log} N)^{d/2}$ for all sequences.

Order of magnitude is optimal for $p eq 1$.

Results hold for arbitrary test intervals in the unit cube.

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.