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

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 ≥ 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 ∈ P , which provides an alternative proof of the known results.