The $L_p$-discrepancy for finite $p>1$ suffers from the curse of dimensionality

Abstract

The $L_p$-discrepancy is a classical quantitative measure for the irregularity of distribution of an $N$-element point set in the $d$-dimensional unit cube. Its inverse for dimension $d$ and error threshold $\varepsilon \in (0,1)$ is the number of points in $[0,1)^d$ that is required such that the minimal normalized $L_p$-discrepancy is less or equal $\varepsilon$. It is well known, that the inverse of $L_2$-discrepancy grows exponentially fast with the dimension $d$, i.e., we have the curse of dimensionality, whereas the inverse of $L_{\infty}$-discrepancy depends exactly linearly on $d$. The behavior of inverse of $L_p$-discrepancy for general $p \not\in \{2,\infty\}$ was an open problem since many years. Recently, the curse of dimensionality for the $L_p$-discrepancy was shown for an infinite sequence of values $p$ in $(1,2]$, but the general result seemed to be out of reach. In the present paper we show that the $L_p$-discrepancy suffers from the curse of dimensionality for all $p$ in $(1,\infty)$ and only the case $p=1$ is still open. This result follows from a more general result that we show for the worst-case error of positive quadrature formulas for an anchored Sobolev space of once differentiable functions in each variable whose first mixed derivative has finite $L_q$-norm, where $q$ is the Hölder conjugate of $p$.

Figures (3)

• Figure 1: Plot of $C_p$ for $p \in [1,50]$. Note that $C_1=1$ and $\lim_{p \rightarrow \infty} C_p=1$. We have $C_2=1.0022\ldots$, $C_3=1.00248\ldots$, $C_4=1.00238\ldots$.
• Figure 2: Decomposition functions $h_{1,1}$, $h_{1,2,(0)}$ and $h_{1,2,(1)}$ for $p=3$ and decomposition point $a=1/2$.
• Figure 3: Plot of $\widetilde{C}_p$ compared to $C_p$ for $p \in [1,50]$. Note that $\widetilde{C}_1=C_1=1$.

Theorems & Definitions (4)

• Theorem 1
• Lemma 2
• Theorem 3
• proof