Linear Bounds for Differentiable Limits of Weak Pair Correlation Functions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911626459873280 |
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| author | Weiß, Christian |
| author_facet | Weiß, Christian |
| contents | For $s \geq 0$ and a parameter $0 < β< 1$, the weak pair correlation function $f_{N,β}(s)$ 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 $\lim_{N \to \infty} f_{N,β}(0) = 0$ 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 $f_β(s)=\lim_{N\to\infty} f_{N,β}(s)$ exists for all $s\ge 0$ and is differentiable in a neighborhood of the origin. Under these assumptions, we establish the bounds $2s \le f_β(s) \le f'_β(0)\, s,$ thereby providing general constraints on the limiting function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_24481 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Linear Bounds for Differentiable Limits of Weak Pair Correlation Functions Weiß, Christian Number Theory For $s \geq 0$ and a parameter $0 < β< 1$, the weak pair correlation function $f_{N,β}(s)$ 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 $\lim_{N \to \infty} f_{N,β}(0) = 0$ 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 $f_β(s)=\lim_{N\to\infty} f_{N,β}(s)$ exists for all $s\ge 0$ and is differentiable in a neighborhood of the origin. Under these assumptions, we establish the bounds $2s \le f_β(s) \le f'_β(0)\, s,$ thereby providing general constraints on the limiting function. |
| title | Linear Bounds for Differentiable Limits of Weak Pair Correlation Functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2604.24481 |