Pair correlations of Halton and Niederreiter Sequences are not Poissonian

Niederreiter and Halton sequences are two prominent classes of higher-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 who hypothesized that the Halton sequences are not Poissonian. The proofs rely on a general tool which identifies a specific regularity of a sequence to be sufficient for not having Poissonian pair correlations.

Automated summary

Niederreiter and Halton sequences, widely used in numerical integration methods for their excellent distribution qualities, do not satisfy the stronger property of Poissonian pair correlations despite being uniformly distributed. The proofs rely on a general tool that identifies specific regularity of a sequence as sufficient for not having Poissonian pair correlations.