PreprintCC BY 4.0arXiv
The Llog L-Luxemburg Discrepancy for Nonnegative Weights Suffers from the Curse of Dimensionality
Abstract
We study the normalized discrepancy in the Luxemburg norm generated by Φ(s)=slog(1+s). For arbitrary nonnegative quadrature weights, we prove that the associated information complexity suffers from the curse of dimensionality: for every ε∈(0,1), there exist constants cε>0 and d0=d0(ε)∈ℕ such that NΦ,+(ε,d)≥ cε d−1(1.07735…)d for all d≥ d0. The result therefore also applies to equal-weight quasi-Monte Carlo rules.
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BibTeX
@misc{dick2026llog,
title = {{The $L\log L$-Luxemburg Discrepancy for Nonnegative Weights Suffers from the Curse of Dimensionality}},
author = {Josef Dick and Friedrich Pillichshammer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21717},
eprint = {2610.21717},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21717},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Llog L-Luxemburg Discrepancy for Nonnegative Weights Suffers from the Curse of Dimensionality
AU - Josef Dick
AU - Friedrich Pillichshammer
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21717
UR - https://arxiv.org/abs/2610.21717
AB - We study the normalized discrepancy in the Luxemburg norm generated by Φ(s)=slog(1+s). For arbitrary nonnegative quadrature weights, we prove that the associated information complexity suffers from the curse of dimensionality: for every ε∈(0,1), there exist constants c_ε>0 and d_0=d_0(ε)∈ℕ such that N_Φ,+(ε,d)≥ c_ε d^−1(1.07735…)^d for all d≥ d_0. The result therefore also applies to equal-weight quasi-Monte Carlo rules.
ER -