The $L_1$-Discrepancy with Nonnegative Weights Suffers from the Curse of Dimensionality

Abstract

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 \[ N_{1,+}(\varepsilon, d) \ge \frac{(1-\varepsilon)^2}{1 + \varepsilon} \left( \frac{3+2 \sqrt{3}}{6}\right)^d, \] where $(3+2\sqrt{3})/6 = 1.07735\ldots$. The proof combines a change to a volume-biased probability measure with a fractional-moment estimate for the normalized discrepancy function. The lower bound applies, in particular, to equally weighted point sets. The argument uses the nonnegativity of the weights in an essential way and does not cover arbitrary signed weights.