Linear Bounds for Differentiable Limits of Weak Pair Correlation Functions

TL;DR

This paper establishes linear bounds for the differentiable limits of weak pair correlation functions in sequences, providing constraints on their behavior near zero.

Contribution

It introduces bounds for the limiting function of weak pair correlation functions assuming differentiability, advancing understanding of their asymptotic behavior.

Findings

The limiting function is bounded below by 2s.

The limiting function is bounded above by its derivative at zero times s.

These bounds hold under differentiability assumptions near zero.

Abstract

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.