A generalized Faulhaber inequality, improved bracketing covers, and applications to discrepancy

Gnewuch, Michael; Pasing, Hendrik; Weiß, Christian

doi:10.1090/mcom/3666

A generalized Faulhaber inequality, improved bracketing covers, and applications to discrepancy
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by Michael Gnewuch, Hendrik Pasing and Christian Weiß;

Math. Comp. 90 (2021), 2873-2898

DOI: https://doi.org/10.1090/mcom/3666

Published electronically: July 15, 2021

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Abstract:

We prove a generalized Faulhaber inequality to bound the sums of the $j$-th powers of the first $n$ (possibly shifted) natural numbers. With the help of this inequality we are able to improve the known bounds for bracketing numbers of $d$-dimensional axis-parallel boxes anchored in $0$ (or, put differently, of lower left orthants intersected with the $d$-dimensional unit cube $[0,1]^d$). We use these improved bracketing numbers to establish new bounds for the star-discrepancy of negatively dependent random point sets and its expectation. We apply our findings also to the weighted star-discrepancy.

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