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On the quasi-uniformity properties of quasi-Monte Carlo lattice point sets and sequences

Josef Dick, Takashi Goda, Gerhard Larcher, Friedrich Pillichshammer, Kosuke Suzuki

TL;DR

This work clarifies the relationship between quasi-uniformity and uniform distribution for lattice-based point sets and sequences, showing that quasi-uniformity demands optimal scaling of both covering and separation radii, while low discrepancy captures uniformity of distribution. It develops explicit constructions (Frolov admissible lattices, rank-1 lattices, Fibonacci lattices) that are simultaneously quasi-uniform and low-discrepancy, and proves an exact characterization for when $(n\boldsymbol{\alpha})$-sequences are quasi-uniform, linking it to Diophantine approximation via badly approximable vectors. The results yield nested, shrinking, shifted lattices with strong space-filling properties and explicit discrepancy bounds, with practical implications for quasi-Monte Carlo integration, radial basis function methods, and kernel-based approximation. In particular, the paper provides concrete examples where both low discrepancy and quasi-uniformity hold, enabling robust, well-spaced node sets for high-dimensional numerical 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. Specifically, we analyze rank-1 lattice point sets, Fibonacci lattice point sets, Frolov point sets, and $(n \boldsymbolα)$-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.

On the quasi-uniformity properties of quasi-Monte Carlo lattice point sets and sequences

TL;DR

This work clarifies the relationship between quasi-uniformity and uniform distribution for lattice-based point sets and sequences, showing that quasi-uniformity demands optimal scaling of both covering and separation radii, while low discrepancy captures uniformity of distribution. It develops explicit constructions (Frolov admissible lattices, rank-1 lattices, Fibonacci lattices) that are simultaneously quasi-uniform and low-discrepancy, and proves an exact characterization for when -sequences are quasi-uniform, linking it to Diophantine approximation via badly approximable vectors. The results yield nested, shrinking, shifted lattices with strong space-filling properties and explicit discrepancy bounds, with practical implications for quasi-Monte Carlo integration, radial basis function methods, and kernel-based approximation. In particular, the paper provides concrete examples where both low discrepancy and quasi-uniformity hold, enabling robust, well-spaced node sets for high-dimensional numerical 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. Specifically, we analyze rank-1 lattice point sets, Fibonacci lattice point sets, Frolov point sets, and -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 -sequence with for is quasi-uniform and has low-discrepancy.

Paper Structure

This paper contains 22 sections, 12 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Quasi-Uniformity
  3. Brief summary of known results.
  4. Uniform distribution modulo 1 and discrepancy
  5. Brief summary of known results.
  6. Quasi-uniformity versus uniform distribution
  7. Quasi-uniformity does not imply uniform distribution.
  8. A sequence of uniformly distributed lattice point sets which are not quasi-uniform.
  9. A quasi-uniform sequence of lattice point sets is always uniformly distributed.
  10. A quasi-uniform sequence of lattice point sets with slowly converging discrepancy.
  11. Motivation and Applications
  12. Organization of the paper
  13. Background on lattices
  14. Successive minima.
  15. Lattice points inside the hypercube.
  16. ...and 7 more sections

Key Result

Lemma 1.3

Let $\mathcal{S} = (\boldsymbol{x}_n)_{n \ge 0}$ be a sequence in $[0,1]^d$. Let $P_i = \{ \boldsymbol{x}_0, \boldsymbol{x}_1, \ldots, \boldsymbol{x}_{i-1}\}$ denote the set of the first $i$ points of $\mathcal{S}$. Let $1 \le i_1 < i_2 < i_3 < \dots$ be an increasing sequence.

Theorems & Definitions (31)

  • Definition 1.1
  • Definition 1.2
  • Lemma 1.3
  • Definition 2.1: Lattices
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 21 more