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arXiv · 2501.18226

On the quasi-uniformity properties of quasi-Monte Carlo point sets and sequences -- Part II: digital nets and sequences

Josef Dick·Takashi Goda·Kosuke Suzuki
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Abstract

We study the quasi-uniformity properties of digital nets, a class of quasi-Monte Carlo point sets. Quasi-uniformity is a space-filling property used for instance in experimental designs and radial basis function approximation. However, it has not been investigated so far whether common low-discrepancy digital nets are quasi-uniform, with the exception of the two-dimensional Sobol' sequence, which has recently been shown not to be quasi-uniform. In this paper, with the goal of constructing quasi-uniform low-discrepancy digital nets, we introduce the notion of well-separated point sets and provide an algebraic criterion to determine whether a given sequence of digital nets is well-separated. Using this criterion, we present an example of a two-dimensional digital net which has low-discrepancy and is quasi-uniform. Additionally, we provide several counterexamples of low-discrepancy digital nets that are not quasi-uniform.

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Josef Dick, Takashi Goda, Kosuke Suzuki. 2025-01-30. On the quasi-uniformity properties of quasi-Monte Carlo point sets and sequences -- Part II: digital nets and sequences. https://arxiv.org/abs/2501.18226

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