2025/02/10 by Josef Dick, Takashi Goda, Dick, Josef +7 · 4 citations
Mathematics · #11J71 #11K36 #11K38 #FOS: Mathematics #Mathematical Approximation and Integration #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2502.06202
openalex publication_date 2025/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
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 αj = 2j/(d+1) for j ∈ \1, 2, …, 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.