I was doing some reading and came across the term "Pair-Correlation Statistic" of a set. They mentioned how this was a statistic that was used to classify spacings of certain sequences, such as the set of Zeroes of the Zeta function, the set of Random Hermitian Matrices. The paper that I am reading discusses the correlations of Farey Fractions. Below is the formal definition that they are using for v-level correlations, and then the 2-level correlation, which is just the pair correlation.
Let $v$ ≥ 1 be an integer and let F be a finite set of N elements in [0, 1]. The $v$-level correlation measure $R^{(ν)}_F(B)$ of a box $B ⊂ \mathbb{R}^{ν−1}$ is defined as
$\frac{1}{N}$#$\{(x_1,...,x_v)\in F^v: x_i $ distinct $, (x_1 −x_2, x_2 −x_3, . . . , x_{ν−1} −x_v) \in \frac{1}{N}B+\mathbb{Z}^{v-1} \}$
When $v=2$, the pair correlation measure of an interval $I \subset \mathbb{R}$ is defined as
$R^{(2)}_F(I) = \frac{1}{N}$#$\{(x,y)\in F^2: x\neq y, x-y \in \frac{1}{N}I + \mathbb{Z}\}$
Intuitively, I understand this as the normalized relative spacing between any 2 unordered pairs of elements in a set. Or how often pairs of differences are contained in an interval. I'm wondering whether this interpretation is correct.
I'm also wondering what this measure would represent, especially considering the Farey Fractions, for $v$-level of arbitrary $v$. For example, what would the $3$ and $4$-level correlation measures mean?
I'd appreciate any help.