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

TL;DR

This paper extends the concept of Poissonian pair correlation to higher dimensions using supremum-norm, explores its relation to uniform distribution, and shows most low-discrepancy sequences lack this property.

Contribution

It introduces a multi-dimensional Poissonian pair correlation concept and analyzes its prevalence among typical low-discrepancy sequences.

Findings

Almost all sequences satisfy the new multi-dimensional Poissonian pair correlation concept.

Most low-discrepancy sequences in high dimensions do not exhibit Poissonian pair correlations.

The new concept links to uniform distribution and is supported by metrical pair correlation theory.

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 $1/N^{\frac{1}{d}}$, 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.