The inverse of the star-discrepancy problem and the generation of pseudo-random numbers

TL;DR

This paper reviews the inverse star-discrepancy problem, which seeks explicit point sets with low discrepancy for high-dimensional integration, and explores its connections to pseudo-random number generation.

Contribution

It summarizes the current state of the inverse star-discrepancy problem and discusses potential links to pseudo-random number generators.

Findings

Existence of low-discrepancy point sets proven since 2001

Open problem: explicit constructions of such point sets

Connections to pseudo-random number generators explored

Abstract

The inverse of the star-discrepancy problem asks for point sets $P_{N,s}$ of size $N$ in the $s$-dimensional unit cube $[0,1]^s$ whose star-discrepancy $D^\ast(P_{N,s})$ satisfies $$D^\ast(P_{N,s}) \le C \sqrt{s/N},$$ where $C> 0$ is a constant independent of $N$ and $s$. The first existence results in this direction were shown by Heinrich, Novak, Wasilkowski, and Wo\'{z}niakowski in 2001, and a number of improvements have been shown since then. Until now only proofs that such point sets exist are known. Since such point sets would be useful in applications, the big open problem is to find explicit constructions of suitable point sets $P_{N,s}$. We review the current state of the art on this problem and point out some connections to pseudo-random number generators.