On the quasi-uniformity properties of quasi-Monte Carlo point sets and sequences -- Part II: digital nets and sequences
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arXiv
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| Format: | Preprint |
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2025
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| author | Dick, Josef Goda, Takashi Suzuki, Kosuke |
| author_facet | Dick, Josef Goda, Takashi Suzuki, Kosuke |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18226 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the quasi-uniformity properties of quasi-Monte Carlo point sets and sequences -- Part II: digital nets and sequences Dick, Josef Goda, Takashi Suzuki, Kosuke Number Theory 05B40, 11K36, 11K45, 52C15, 52C17 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. |
| title | On the quasi-uniformity properties of quasi-Monte Carlo point sets and sequences -- Part II: digital nets and sequences |
| topic | Number Theory 05B40, 11K36, 11K45, 52C15, 52C17 |
| url | https://arxiv.org/abs/2501.18226 |