Exact $\ell^\infty$-separation radius of Sobol' sequences in dimension 2

TL;DR

This paper derives exact formulas for the $ ext{l}^ ext{infty}$-separation radius of the first $N=2^m$ points in the 2D Sobol' sequence, revealing it is suboptimal compared to the ideal rate and grows at least as $N^{1/4}$.

Contribution

It provides the exact $ ext{l}^ ext{infty}$-separation radius for all $N=2^m$ in 2D Sobol' sequences, clarifying their quasi-uniformity properties.

Findings

Separation radius is $O(N^{-3/4})$ for Sobol' points.

Sobol' sequence has a mesh ratio growing at least as $N^{1/4}$.

Exact formulas for the separation radius are established.

Abstract

Quasi-uniformity is a fundamental geometric property of point sets, crucial for applications such as kernel interpolation, Gaussian process regression, and space-filling experimental designs. While quasi-Monte Carlo methods are widely recognized for their low-discrepancy characteristics, understanding their quasi-uniformity remains important for practical applications. For the two-dimensional Sobol' sequence, Sobol' and Shukhman (2007) conjectured that the separation radius of the first $N$ points achieves the optimal rate $N^{-1/2}$, which would imply quasi-uniformity. This conjecture was disproved by Goda (2024), who computed exact values of the $\ell^2$-separation radius for a sparse subsequence $N = 2^{2^v-1}$. However, the general behavior of the Sobol' sequence for arbitrary $N$ remained unclear. In this paper, we derive exact expressions for the $\ell^\infty$-separation radius of the first $N = 2^m$ points of the two-dimensional Sobol' sequence for all $m \in \mathbb{N}$. As an immediate consequence, we show that the separation radius of Sobol' points is $O(N^{-3/4})$, which is strictly worse than the optimal rate $N^{-1/2}$, revealing that the two-dimensional Sobol' sequence has a suboptimal mesh ratio that grows at least as $N^{1/4}$.