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 minimal number of points in $[0,1)^d$ 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 $p \not\in \{2,\infty\}$ has been an open problem for many years. In this paper we show that the $L_p$-discrepancy suffers from the curse of dimensionality for all $p$ in $(1,2]$ 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：$L_p$-偏差是衡量$d$维单位立方体中$N$个点集分布不均匀性的定量指标，它与数值积分的拟蒙特卡洛方法的误差上限密切相关。对于维度$d$和误差阈值$\varepsilon \in (0,1)$，其逆是$[0,1)^d$中使最小归一化$L_p$-偏差不超过$\varepsilon$所需的最小点数。众所周知，$L_2$-偏差的逆随维度$d$呈指数增长，即存在维数灾难，而$L_{\infty}$-偏差的逆对$d$是严格线性的。对于一般$p \not\in \{2,\infty\}$的$L_p$-偏差的逆的行为多年来一直是一个开放问题。本文证明，对于所有形如$p=2 \ell/(2 \ell -1)$（$\ell \in \mathbb{N}$）且位于$(1,2]$区间的$p$，$L_p$-偏差存在维数灾难。这一结果源于一个更一般的结论：我们证明了在锚点为0的锚定Sobolev空间中，对于每个变量上具有一次可微性且一阶导数具有有限$L_q$-范数的函数（其中$q$为满足$1/p+1/q=1$的偶正整数），数值积分的误差上限存在维数灾难。