Exact $\ell^\infty$-separation radius of Sobol' sequences in dimension 2
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| Format: | Preprint |
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2025
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| _version_ | 1866911334182944768 |
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| author | Suzuki, Kosuke |
| author_facet | Suzuki, Kosuke |
| contents | Quasi-uniformity is a fundamental geometric property of point sets, crucial for applications such as kernel interpolation, Gaussian process regression, and space-filling experimental designs. While quasi-Monte Carlo methods are widely recognized for their low-discrepancy characteristics, understanding their quasi-uniformity remains important for practical applications. For the two-dimensional Sobol' sequence, Sobol' and Shukhman (2007) conjectured that the separation radius of the first $N$ points achieves the optimal rate $N^{-1/2}$, which would imply quasi-uniformity. This conjecture was disproved by Goda (2024), who computed exact values of the $\ell^2$-separation radius for a sparse subsequence $N = 2^{2^v-1}$. However, the general behavior of the Sobol' sequence for arbitrary $N$ remained unclear. In this paper, we derive exact expressions for the $\ell^\infty$-separation radius of the first $N = 2^m$ points of the two-dimensional Sobol' sequence for all $m \in \mathbb{N}$. As an immediate consequence, we show that the separation radius of Sobol' points is $O(N^{-3/4})$, which is strictly worse than the optimal rate $N^{-1/2}$, revealing that the two-dimensional Sobol' sequence has a suboptimal mesh ratio that grows at least as $N^{1/4}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_14803 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exact $\ell^\infty$-separation radius of Sobol' sequences in dimension 2 Suzuki, Kosuke Numerical Analysis 05B40, 11K36, 52C15 Quasi-uniformity is a fundamental geometric property of point sets, crucial for applications such as kernel interpolation, Gaussian process regression, and space-filling experimental designs. While quasi-Monte Carlo methods are widely recognized for their low-discrepancy characteristics, understanding their quasi-uniformity remains important for practical applications. For the two-dimensional Sobol' sequence, Sobol' and Shukhman (2007) conjectured that the separation radius of the first $N$ points achieves the optimal rate $N^{-1/2}$, which would imply quasi-uniformity. This conjecture was disproved by Goda (2024), who computed exact values of the $\ell^2$-separation radius for a sparse subsequence $N = 2^{2^v-1}$. However, the general behavior of the Sobol' sequence for arbitrary $N$ remained unclear. In this paper, we derive exact expressions for the $\ell^\infty$-separation radius of the first $N = 2^m$ points of the two-dimensional Sobol' sequence for all $m \in \mathbb{N}$. As an immediate consequence, we show that the separation radius of Sobol' points is $O(N^{-3/4})$, which is strictly worse than the optimal rate $N^{-1/2}$, revealing that the two-dimensional Sobol' sequence has a suboptimal mesh ratio that grows at least as $N^{1/4}$. |
| title | Exact $\ell^\infty$-separation radius of Sobol' sequences in dimension 2 |
| topic | Numerical Analysis 05B40, 11K36, 52C15 |
| url | https://arxiv.org/abs/2508.14803 |