The $L\log L$-Luxemburg Discrepancy for Nonnegative Weights Suffers from the Curse of Dimensionality

Abstract

We study the normalized discrepancy in the Luxemburg norm generated by $\Phi(s)=s\log(1+s)$. 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 \[N_{\Phi,+}(\varepsilon,d)\ge c_{\varepsilon} d^{-1}(1.07735\ldots)^d \qquad \mbox{for all } d\ge d_0. \] The result therefore also applies to equal-weight quasi-Monte Carlo rules.