Pair Correlations in Uniform Countable Sets

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.