On the quasi-uniformity properties of quasi-Monte Carlo point sets and sequences -- Part I: Lattices and Kronecker sequences
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arXiv
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2025
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| author | Dick, Josef Goda, Takashi Larcher, Gerhard Pillichshammer, Friedrich Suzuki, Kosuke |
| author_facet | Dick, Josef Goda, Takashi Larcher, Gerhard Pillichshammer, Friedrich Suzuki, Kosuke |
| contents | The discrepancy of a point set quantifies how well the points are distributed, with low-discrepancy point sets demonstrating exceptional uniform distribution properties. Such sets are integral to quasi-Monte Carlo methods, which approximate integrals over the unit cube for integrands of bounded variation. In contrast, quasi-uniform point sets are characterized by optimal separation and covering radii, making them well-suited for applications such as radial basis function approximation. This paper explores the quasi-uniformity properties of quasi-Monte Carlo point sets constructed from lattices and also Kronecker sequences. Specifically, we analyze rank-1 lattice point sets, Fibonacci lattice point sets, Frolov point sets, and Kronecker sequences (also referred to as $(n \boldsymbolα)$-sequences), providing insights into their potential for use in applications that require both low-discrepancy and quasi-uniform distribution. As an example, we show that the $(n \boldsymbolα)$-sequence with $α_j = 2^{j/(d+1)}$ for $j \in \{1, 2, \ldots, d\}$ is quasi-uniform and has low-discrepancy. The quasi-uniformity properties of quasi-Monte Carlo digital nets and sequences will be studied in a companion paper. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_06202 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the quasi-uniformity properties of quasi-Monte Carlo point sets and sequences -- Part I: Lattices and Kronecker sequences Dick, Josef Goda, Takashi Larcher, Gerhard Pillichshammer, Friedrich Suzuki, Kosuke Number Theory 11K36, 11K38, 11J71 The discrepancy of a point set quantifies how well the points are distributed, with low-discrepancy point sets demonstrating exceptional uniform distribution properties. Such sets are integral to quasi-Monte Carlo methods, which approximate integrals over the unit cube for integrands of bounded variation. In contrast, quasi-uniform point sets are characterized by optimal separation and covering radii, making them well-suited for applications such as radial basis function approximation. This paper explores the quasi-uniformity properties of quasi-Monte Carlo point sets constructed from lattices and also Kronecker sequences. Specifically, we analyze rank-1 lattice point sets, Fibonacci lattice point sets, Frolov point sets, and Kronecker sequences (also referred to as $(n \boldsymbolα)$-sequences), providing insights into their potential for use in applications that require both low-discrepancy and quasi-uniform distribution. As an example, we show that the $(n \boldsymbolα)$-sequence with $α_j = 2^{j/(d+1)}$ for $j \in \{1, 2, \ldots, d\}$ is quasi-uniform and has low-discrepancy. The quasi-uniformity properties of quasi-Monte Carlo digital nets and sequences will be studied in a companion paper. |
| title | On the quasi-uniformity properties of quasi-Monte Carlo point sets and sequences -- Part I: Lattices and Kronecker sequences |
| topic | Number Theory 11K36, 11K38, 11J71 |
| url | https://arxiv.org/abs/2502.06202 |