Disproving the quasi-uniformity of the Halton sequences and of some Halton-type sequences

TL;DR

This paper demonstrates that the Halton sequence and certain Halton-type sequences are not quasi-uniform in dimensions two and higher, challenging assumptions about their uniformity properties.

Contribution

It provides a rigorous proof that the Halton sequence and some Halton-type sequences lack quasi-uniformity in higher dimensions, offering an alternative proof for known results.

Findings

Halton sequence is not quasi-uniform in any dimension d ≥ 2

Certain Halton-type sequences, including Faure sequences, are not quasi-uniform

Provides an alternative proof of existing non-quasi-uniformity results

Abstract

In this short article, we prove that the Halton sequence, one of the most well-known low-discrepancy sequences, is not quasi-uniform in any dimension $d \ge 2$ with any pairwise relatively prime bases. We further disprove the quasi-uniformity of some Halton-type sequences, including the $p$-dimensional Faure sequence in base $p$, $p \in \mathbb{P}$, which provides an alternative proof of the known results.