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The Sobol' sequence is not quasi-uniform in dimension 2

Takashi Goda

TL;DR

The paper addresses whether the 2D Sobol' sequence is quasi-uniform by examining the mesh ratio defined via the fill distance $h(P_n)$ and separation radius $q(P_n)$. It leverages the explicit generating matrices for the 2D Sobol' sequence, notably the Pascal matrix modulo $2$, and employs Lucas' theorem to establish parity properties that yield explicit lower bounds. The main result shows there exist infinitely many $n$ with $q(P_n)=1/(\sqrt{2}(n+1))$, thereby proving the 2D Sobol' sequence is not quasi-uniform; this settles the question in dimension $2$ and leaves higher dimensions as future work. The finding narrows the understanding of quasi-uniformity for common QMC point sets and has implications for their use in kernel interpolation and Gaussian process regression, where mesh ratio behavior informs error bounds.

Abstract

Are common quasi-Monte Carlo sequences quasi-uniform? While this question remains widely open, in this short note, we prove that the two-dimensional Sobol' sequence is not quasi-uniform. This result partially answers an unsolved problem of Sobol' and Shukhman (2007) in a negative manner.

The Sobol' sequence is not quasi-uniform in dimension 2

TL;DR

The paper addresses whether the 2D Sobol' sequence is quasi-uniform by examining the mesh ratio defined via the fill distance and separation radius . It leverages the explicit generating matrices for the 2D Sobol' sequence, notably the Pascal matrix modulo , and employs Lucas' theorem to establish parity properties that yield explicit lower bounds. The main result shows there exist infinitely many with , thereby proving the 2D Sobol' sequence is not quasi-uniform; this settles the question in dimension and leaves higher dimensions as future work. The finding narrows the understanding of quasi-uniformity for common QMC point sets and has implications for their use in kernel interpolation and Gaussian process regression, where mesh ratio behavior informs error bounds.

Abstract

Are common quasi-Monte Carlo sequences quasi-uniform? While this question remains widely open, in this short note, we prove that the two-dimensional Sobol' sequence is not quasi-uniform. This result partially answers an unsolved problem of Sobol' and Shukhman (2007) in a negative manner.
Paper Structure (2 sections, 2 theorems, 21 equations)

This paper contains 2 sections, 2 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Results

Key Result

Lemma 2.1

Let $m=2^{w}-1$ for an integer $w\geq 1$. For any $1\leq i\leq m$, it holds that

Theorems & Definitions (4)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof