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 $\epsilon \in (0,1)$ is the number of points in $[0,1{)}^d$ that is required such that the minimal normalized $L_p$-discrepancy is less or equal $\epsilon$. 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 $p\notin \{2,\infty \}$ 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 $p$.

arXiv admin note: substantial text overlap with arXiv:2303.01787