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Hidden low-discrepancy structures in random point sets

Kohei Suzuki, Takashi Goda

TL;DR

This paper addresses whether a purely random point set in $[0,1)^d$ must contain a low-discrepancy $(0,m,d)$-net in base $b$. It develops a counting bound on admissible net patterns $a_{b,d}(m)$ and applies a second-moment analysis (Paley–Zygmund) to link pattern existence to sampling size $N$. The main result identifies scaling thresholds: if $N \ge (1+\varepsilon) b^{md} m \log b$, the probability tends to 1; necessary conditions are also derived. This work bridges randomness and structured QMC constructs, revealing how high-structure subsets emerge in random clouds and guiding strategies for sampling and downsampling in high dimensions.

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 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 concentration inequalities together with this bound, we give lower and upper estimates for the probability that a set of $N$ random points contains a $(0, m, d)$-net as a subset. This result leads to necessary and sufficient scaling conditions on $N$ and $m$ such that this probability converges to $1$.

Hidden low-discrepancy structures in random point sets

TL;DR

This paper addresses whether a purely random point set in must contain a low-discrepancy -net in base . It develops a counting bound on admissible net patterns and applies a second-moment analysis (Paley–Zygmund) to link pattern existence to sampling size . The main result identifies scaling thresholds: if , the probability tends to 1; necessary conditions are also derived. This work bridges randomness and structured QMC constructs, revealing how high-structure subsets emerge in random clouds and guiding strategies for sampling and downsampling in high dimensions.

Abstract

We study the probabilistic existence of point configurations satisfying the -net property in base within a randomly generated point set in the -dimensional unit cube. We first derive an upper bound on the number of geometric patterns for -nets in base . By applying the concentration inequalities together with this bound, we give lower and upper estimates for the probability that a set of random points contains a -net as a subset. This result leads to necessary and sufficient scaling conditions on and such that this probability converges to .

Paper Structure

This paper contains 3 sections, 2 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results

Key Result

Theorem 3.1

For $d\in \mathbb N$, let $\mathcal{S}_N$ be a set of $N$ independently and uniformly distributed random points in $[0, 1)^d$. Let $b,m\ge 2$ such that $b$ is a prime-power with $d\le b+1$. Let $C_{b,d}(N,m)$ denote the event that $\mathcal{S}_N$ contains a $(0,m,d)$-net in base $b$. To ensure the c for some constant $\epsilon > 0$, whereas the necessary condition is

Theorems & Definitions (6)

  • Definition 2.1
  • Remark 2.2
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}