On a multi-dimensional Poissonian pair correlation concept and uniform distribution

Abstract

The aim of the present article is to introduce a concept which allows to generalise the notion of Poissonian pair correlation, a second-order equidistribution property, to higher dimensions. Roughly speaking, in the one-dimensional setting, the pair correlation statistics measures the distribution of spacings between sequence elements in the unit interval at distances of order of the mean spacing 1/N . In the d-dimensional case, of course, the order of the mean spacing is 1/N 1 d , and -in our concept-the distance of sequence elements will be measured by the supremum-norm.Additionally, we show that, in some sense, almost all sequences satisfy this new concept and we examine the link to uniform distribution. The metrical pair correlation theory is investigated and it is proven that a class of typical low-discrepancy sequences in the high-dimensional unit cube do not have Poissonian pair correlations, which fits the existing results in the one-dimensional case.

Pair correlation and equidistribution on manifolds

“…The answer is yes, even under weaker hypotheses than (1.2), for sequences in the unit interval [1,8,18]. The same has been established for point sequences on higher-dimensional tori [10,19]. In the present paper we develop a statistical argument that permits a generalisation of these findings to bounded domains in R d (Section 2) as well as compact Riemannian manifolds (Section 3; the special case of flat tori is discussed in the appendix).…”

7. On pair correlation of sequences

“…Of course it makes sense to generalize the concept of PPC to the multidimensional setting. One way to generalize the one-dimensional concept to a multi-dimensioanl setting was defined and discussed in [15] (for a more general analysis of a multi-dimensional PPC concept, we refer to the recent work [21]). Here, we present the definition of [15].…”

Pair correlations of Halton and Niederreiter Sequences are not Poissonian

“…A generalization of Poissonian pair correlations to a higher-dimensional setting has recently been established in [ 9 ] as well as in [ 22 ] (see also [ 16 ] for a more general analysis of higher-dimensional pair correlations). In this work, we refer to the concept in [ 9 ]. To this end, let denote a combination of the supremum-norm of a d -dimensional vector with the distance to the nearest integer function defined by A d -dimensional sequence has Poissonian pair correlations if for every real number as .…”

“…The motivation behind the present paper was to prove that both the Halton sequences and the Niederreiter sequences do not have Poissonian pair correlations in the sense of ( 3 ). We emphasize that it is not ad hoc clear whether the two higher-dimensional concepts of Poissonian pair correlations in [ 9 ] and [ 22 ] are equivalent or not. Therefore the first task for the investigation is to choose one of the concepts in higher dimensions.…”

“…[ 3 ] for more details. The question whether Halton sequences have Poissonian pair correlations or not was posed in [ 9 ] and also stated as Problem 5 in [ 13 ], although it was suggested that this is most likely not the case. It turns out that this conjecture indeed is true.…”