Josef Dick

The L1-Discrepancy with Nonnegative Weights Suffers from the Curse of Dimensionality

We prove that the $L_1$-discrepancy with arbitrary nonnegative weights suffers from the curse of dimensionality. More precisely, for every $\varepsilon \in (0,1)$ and $d \in \mathbb{N}$, the inverse of the $L_1$-discrepancy satisfies \[ N_{1,+}(\varepsilon, d) \ge \frac{(1-\varepsilon)^2}{1 + \varepsilon} \left( \frac{...