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

2025/11/26 by Takashi Goda, Goda, Takashi, Roswitha Hofer +3
Mathematics · #Mathematical Approximation and Integration #Analytic Number Theory Research #Advanced Harmonic Analysis Research

paper · pdf · doi:10.48550/arxiv.2511.21153

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 ≥ 2 with any pairwise relatively prime bases. We further disprove the quasi-uniformity of some Halton-type sequences, including the p-dimensional Faure sequence in base p, p ∈ ℙ, which provides an alternative proof of the known results.

Citations

• Exact ℓ^∞-separation radius of Sobol' sequences in dimension 2
• On the quasi-uniformity properties of quasi-Monte Carlo lattice point sets and sequences
• One-dimensional quasi-uniform Kronecker sequences
• The Sobol' sequence is not quasi-uniform in dimension 2
• On multiscale quasi-interpolation of scattered scalar- and manifold-valued functions
• Convergence Guarantees for Gaussian Process Means With Misspecified\n Likelihoods and Smoothness
• Tractability properties of the weighted star discrepancy of the Halton sequence
• On the lower bound of the discrepancy of Halton's sequence II
• On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals

Related

• A lower bound on the star discrepancy of generalized Halton sequences in rational bases
• Scrambled Halton Subsequences and Inverse Star-Discrepancy
• A metric result for special subsequences of the Halton sequences
• Tractability properties of the weighted star discrepancy of the Halton sequence
• On the lower bound of the discrepancy of Halton's sequence II