On the quasi-uniformity properties of quasi-Monte Carlo point sets and sequences -- Part I: Lattices and Kronecker sequences

TL;DR

This paper investigates the quasi-uniformity and discrepancy properties of specific quasi-Monte Carlo point sets, such as lattices and Kronecker sequences, highlighting their potential for applications requiring uniform distribution and low discrepancy.

Contribution

It provides a detailed analysis of the quasi-uniformity of lattice-based and Kronecker sequences, including explicit examples demonstrating their properties.

Findings

Kronecker sequences with specific parameters are quasi-uniform and low-discrepancy.

Lattice and Fibonacci point sets exhibit favorable distribution properties.

The study offers insights into their suitability for numerical integration and approximation tasks.

Abstract

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 $(n \boldsymbol{\alpha})$-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 $(n \boldsymbol{\alpha})$-sequence with $\alpha_j = 2^{j/(d+1)}$ 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.