Linear Bounds for Differentiable Limits of Weak Pair Correlation Functions

For $s \geq 0$ and a parameter $0 < \beta < 1$, the weak pair correlation function $f_{N,\beta}(s)$ for the first $N \in \mathbb{N}$ elements of a sequence $(x_n)_{n \in \mathbb{N}} \subset[0,1]$ is evidently non-decreasing in $s$. Moreover, it satisfies $\lim_{N \to \infty} f_{N,\beta}(0) = 0$ if the elements of $(x_n)_{n \in \mathbb{N}}$ are distinct. Beyond these basic observations, little is known in general about the behavior of the limiting function. In this note, we investigate the situation in which the limit $f_\beta(s)=\lim_{N\to\infty} f_{N,\beta}(s)$ exists for all $s\ge 0$ and is differentiable in a neighborhood of the origin. Under these assumptions, we establish the bounds $2s \le f_\beta(s) \le f'_\beta(0)\, s,$ thereby providing general constraints on the limiting function.