The curse of dimensionality for the Lp-discrepancy with finite p

Abstract

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. It's inverse for dimension d and error threshold $\epsilon \in (0,1)$ is the minimal number of points in ${[0,1)}^d$ such that the minimal normalized $L_p$-discrepancy is less or equal ε. 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 $p\notin \{2,\infty \}$ 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 $\ell \in N$.

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 $1/p+1/q=1$.