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

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.