Linear Bounds for Differentiable Limits of Weak Pair Correlation Functions

For $s\ge 0$ 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 ${\lim }_{N\to \mathrm{\infty }}{f}_{N,\beta }(0)=0$ 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 $f_{\beta }(s)={\lim }_{N\to \mathrm{\infty }}{f}_{N,\beta }(s)$ exists for all $s\ge 0$ and is differentiable in a neighborhood of the origin. Under these assumptions, we establish the bounds $2s\le f_{\beta }(s)\le f_{\beta }^{\mathrm{\prime }}(0)s,$ thereby providing general constraints on the limiting function.