Abstract
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.
Highlights
Let · denote the distance to the nearest integer
In the last few years pair correlations have been studied from a purely mathematical point of view as the property of Poissonian pair correlations is natural for a sequence of independently chosen random numbers drawn from the uniform distribution
3 Discussion and further research. It is an interesting task for future research to figure out whether the Halton and the Niederreiter sequence do have Poissonian pair correlations in the sense of [22] or not
Summary
Let · denote the distance to the nearest integer. A sequence (xn)n≥0 of real numbers in [0, 1) has Poissonian pair correlations if. In this paper we study the most prominent example of the row-by-row concept by Niederreiter [18] and the most basic form of the column-by-column concept [10] This two classes of digital sequences are generated by non-singular upper triangular (NU T ) matrices and satisfy u = 0 for appropriate e. C(d) are N U T matrices and generate a (0, e, d)-sequence in base q (cf [10, proof of Theorem 1]) For both methods—the Niederreiter construction as well as the column-by-column construction—we strongly focus on analyzing the left hand side of (3) in order to obtain our first main result. C(d) obtained via the Niederreiter construction or the alternative column-by-column approach do not have Poissonian pair correlations Note that both methods of constructing generating matrices based on the specific choice of distinct monic linear polynomials q1(x), . It is a non trivial task to generalize the method of proof of Theorem 1 to more general not necessarily upper triangular generating matrices, i.e. to cover the generalized Niederreiter sequences as well as the Sobol sequences, and the Hofer– Niederreiter sequences [11] and Xing–Niederreiter sequences [26]
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