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Disproving the quasi-uniformity of the Halton sequences and some of Halton-type sequences

Takashi Goda, Roswitha Hofer, Kosuke Suzuki

TL;DR

The paper proves that Halton sequences in dimensions $d\ge 2$ with pairwise coprime bases are not quasi-uniform by establishing that the separation radius $q(P_N)$ can decay faster than the optimal $N^{-1/d}$ along infinitely many $N$, specifically $q(P_N) \lesssim N^{-1/d} (\log N)^{-1/(d(d-1))}$. It introduces a constructive number-theoretic framework to generate close pairs $(n,m)$ across all bases, using a divisibility lemma for $p\equiv 1 \pmod q$ and Dirichlet simultaneous approximation to balance digits across bases. The work extends to Halton-type sequences over polynomial bases in $\mathbb{F}_p[X]$, providing analogous non-quasi-uniformity results in several concrete cases (including Faure-related constructions) and a general covering-radius bound for these digital constructions via a polynomial-analytic approach. These findings clarify the geometric limitations of Halton and Halton-type sequences beyond their discrepancy properties, with implications for their use in numerical integration and scattered-data approximation.

Abstract

In this short article, we prove that the Halton sequence, one of the most well-known low-discrepancy sequences, is not quasi-uniform in any dimension $d \ge 2$ with any pairwise relatively prime bases. We further disprove the quasi-uniformity of some Halton-type sequences, including the Faure sequence, which provides an alternative proof of the known results.

Disproving the quasi-uniformity of the Halton sequences and some of Halton-type sequences

TL;DR

The paper proves that Halton sequences in dimensions with pairwise coprime bases are not quasi-uniform by establishing that the separation radius can decay faster than the optimal along infinitely many , specifically . It introduces a constructive number-theoretic framework to generate close pairs across all bases, using a divisibility lemma for and Dirichlet simultaneous approximation to balance digits across bases. The work extends to Halton-type sequences over polynomial bases in , providing analogous non-quasi-uniformity results in several concrete cases (including Faure-related constructions) and a general covering-radius bound for these digital constructions via a polynomial-analytic approach. These findings clarify the geometric limitations of Halton and Halton-type sequences beyond their discrepancy properties, with implications for their use in numerical integration and scattered-data approximation.

Abstract

In this short article, we prove that the Halton sequence, one of the most well-known low-discrepancy sequences, is not quasi-uniform in any dimension with any pairwise relatively prime bases. We further disprove the quasi-uniformity of some Halton-type sequences, including the Faure sequence, which provides an alternative proof of the known results.

Paper Structure

This paper contains 6 sections, 7 theorems, 62 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some known results
  3. Halton sequences are not quasi-uniform
  4. Further results: Halton-type sequences
  5. Definitions
  6. Results

Key Result

Theorem 1

Let $d \ge 2$, and let $b_1, \dots, b_d$ be pairwise relatively prime integers with $1<b_1 < \cdots < b_d$. Let $\mathcal{S}$ be the $d$-dimensional Halton sequence in bases $b_1,\dots,b_d$. Then there exists a constant $c > 0$ such that, for infinitely many $N \in \mathbb N$, we have In particular, $\mathcal{S}$ is not quasi-uniform.

Figures (1)

  • Figure 1: Separation radius of the Halton sequence from $N=2$ to $N=10^5$ in dimensions $2$ (blue), $3$ (red), $4$ (orange), and $5$ (purple). The corresponding reference lines $N^{-1/d}$ are also plotted in the same colors.

Theorems & Definitions (20)

  • Theorem 1
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • proof : Proof of Theorem \ref{['thm:halton_not_quasi-uniform']}
  • Definition 1: $b(X)$-adic radical inverse function $\varphi_{b(X)}$
  • Definition 2: Halton-type sequence
  • ...and 10 more