Pair Correlations in Uniform Countable Sets

TL;DR

This paper analyzes the pair correlations of countable sets in Euclidean space, providing explicit formulas and limits, and verifying conditions for lattice point sets.

Contribution

It introduces a method to compute pair correlations for sets satisfying equidistribution, with explicit formulas for low dimensions and asymptotic behavior.

Findings

Pair correlations expressed via incomplete Beta function.

Closed form formulas for n=2 and n=3.

Verification for lattice and primitive lattice points.

Abstract

We determine the pair correlations of countable sets $T \subset \mathbb{R}^n$ satisfying natural equidistribution conditions. The pair correlations are computed as the volume of a certain region in $\mathbb{R}^{2n}$, which can be expressed in terms of the incomplete Beta function. For $n=2$ and $n=3$ we give closed form expressions, and we obtain an expression in the $n \rightarrow \infty$ limit. We also verify that sets of lattice points and primitive lattice points satisfy the required distribution criteria.