The $L_p$-discrepancy is a quantitative measure for the irregularity of distribution of an $N$-element point set in the $d$-dimensional unit cube, which is closely related to the worst-case error of quasi-Monte Carlo algorithms for numerical integration. 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 $p \not\in \{2,\infty\}$ is an open problem since many years. In this paper we show that the $L_p$-discrepancy suffers from the curse of dimensionality for all $p$ which are of the form $p=2 \ell/(2 \ell -1)$ with $\ell \in \mathbb{N}$. This result follows from a more general result that we show for the worst-case error of numerical integration in an anchored Sobolev space with anchor 0 of once differentiable functions in each variable whose first derivative has finite $L_q$-norm, where $q$ is an even positive integer satisfying $1/p+1/q=1$.

翻译：$L_p$差异是衡量$d$维单位立方体中$N$个点集分布不规则性的定量指标，它与数值积分的拟蒙特卡洛算法的最坏情况误差密切相关。对于维度$d$和误差阈值$\varepsilon \in (0,1)$，其逆问题是指需要$[0,1)^d$中多少个点才能使得最小归一化$L_p$差异小于等于$\varepsilon$。众所周知，$L_2$差异的逆随维度$d$呈指数增长，即存在维数灾难，而$L_{\infty}$差异的逆则严格线性依赖于$d$。对于一般的$p \not\in \{2,\infty\}$，$L_p$差异的逆行为多年来一直是一个开放问题。在本文中，我们证明对于所有形如$p=2 \ell/(2 \ell -1)$且$\ell \in \mathbb{N}$的$p$，$L_p$差异都存在维数灾难。这一结果源于我们针对数值积分最坏情况误差证明的更一般结论：在一个锚点为0的锚定Sobolev空间中，每个变量的一次可微函数的一阶导数具有有限$L_q$范数，其中$q$为满足$1/p+1/q=1$的偶正整数。