On the number of lattice points in thin sectors
On the circle of radius \(R\) centred at the origin, consider a ``thin'' sector about the fixed line \(y = \alpha x\) with edges given by the lines \(y = (\alpha \pm \epsilon) x\), where \(\epsilon = \epsilon_R \rightarrow 0\) as \( R \to \infty \). We establish an asymptotic count for \(S...
Gespeichert in:
Veröffentlicht in: | arXiv.org 2023-05 |
---|---|
Hauptverfasser: | , |
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
Online-Zugang: | Volltext |
Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
Zusammenfassung: | On the circle of radius \(R\) centred at the origin, consider a ``thin'' sector about the fixed line \(y = \alpha x\) with edges given by the lines \(y = (\alpha \pm \epsilon) x\), where \(\epsilon = \epsilon_R \rightarrow 0\) as \( R \to \infty \). We establish an asymptotic count for \(S_{\alpha}(\epsilon,R)\), the number of integer lattice points lying in such a sector. Our results depend both on the decay rate of \(\epsilon\) and on the rationality/irrationality type of \(\alpha\). In particular, we demonstrate that if \(\alpha\) is Diophantine, then \(S_{\alpha}(\epsilon,R)\) is asymptotic to the area of the sector, so long as \(\epsilon R^{t} \rightarrow \infty\) for some \( t |
---|---|
ISSN: | 2331-8422 |