Pair Correlations of Niederreiter and Halton Sequences are not Poissonian

TL;DR

This paper demonstrates that Niederreiter and Halton sequences, despite their uniform distribution, do not exhibit Poissonian pair correlations, confirming a conjecture and extending previous one-dimensional results.

Contribution

It proves that these multi-dimensional sequences lack Poissonian pair correlations, extending known results and confirming a conjecture about their regularity properties.

Findings

Niederreiter and Halton sequences are not Poissonian in pair correlations.

The result extends to multi-dimensional sequences.

Confirms a conjecture of Larcher and Stockinger.

Abstract

Niederreiter and Halton sequences are two prominent classes of multi-dimensional sequences which are widely used in practice for numerical integration methods because of their excellent distribution qualities. In this paper, we show that these sequences - even though they are uniformly distributed - fail to satisfy the stronger property of Poissonian pair correlations. This extends already established results for one-dimensional sequences and confirms a conjecture of Larcher and Stockinger. The proofs rely on a general tool which identifies specific regularities of a sequence to be sufficient for not having Poissonian pair correlations.