The $L\log L$-Luxemburg Discrepancy for Nonnegative Weights Suffers from the Curse of Dimensionality
Year: 2026
Type: preprint
Abstract: We study the normalized discrepancy in the Luxemburg norm generated by $Φ(s)=s\log(1+s)$. For arbitrary nonnegative quadrature weights, we prove that the associated information complexity suffers from the curse of dimensionality: for every $\varepsilon\in(0,1)$, there exist constants $c_\varepsilon>0$ and $d_0=d_0(\varepsilon)\in\mathbb{N}$ such that \[N_{Φ,+}(\varepsilon,d)\ge c_{\varepsilon} d^{-1}(1.07735\ldots)^d \qquad \mbox{for all } d\ge d_0. \] The result therefore also applies to equal-weight quasi-Monte Carlo rules.
Source: arXiv (Cornell University)
Authors: Dick, Josef, Pillichshammer, Friedrich
Institutions: UNSW Sydney, Johannes Kepler University of Linz
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Austrian Science Fund
OpenAlex ID: W7222753969