Hidden low-discrepancy structures in random point sets

TL;DR

This paper investigates the likelihood of random point sets in high-dimensional spaces forming low-discrepancy structures called $(0, m, d)$-nets, providing probabilistic bounds and conditions for their existence.

Contribution

It introduces new probabilistic bounds on the occurrence of $(0, m, d)$-nets in random point sets and establishes scaling conditions for their high-probability existence.

Findings

Derived an upper bound on geometric patterns for $(0, m, d)$-nets

Established probability thresholds for the existence of such nets

Provided conditions under which random point sets almost surely contain these structures

Abstract

We study the probabilistic existence of point configurations satisfying the $(0, m, d)$-net property in base $b$ within a randomly generated point set of size $N$ in the $d$-dimensional unit cube. We first derive an upper bound on the number of geometric patterns for $(0, m, d)$-nets in base $b$. By applying the elementary probability bounds together with this counting result, we then give scaling conditions on $N$ as a function of $m$ such that this probability converges to $1$ and $0$, respectively.