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 \(y = \alpha x\) $y=\alpha x$ with edges given by the lines \(y = (\alpha \pm \epsilon ) x\) $y=(\alpha \pm ϵ)x$ , where \(\epsilon = \epsilon _R \rightarrow 0\) $ϵ={ϵ}_R\to 0$ as \( R \rightarrow \infty \) $R\to \infty$ . We establish an asymptotic count for \(S_{\alpha }(\epsilon ,R)\) $S_{\alpha }(ϵ,R)$ , the number of integer lattice points lying in such a sector. Our results depend both on the decay rate of \(\epsilon \) $ϵ$ and on the rationality/irrationality type of \(\alpha \) $\alpha$ . In particular, we demonstrate that if \(\alpha \) $\alpha$ is Diophantine, then \(S_{\alpha }(\epsilon ,R)\) $S_{\alpha }(ϵ,R)$ is asymptotic to the area of the sector, so long as \(\epsilon R^{t} \rightarrow \infty \) $ϵR^t\to \infty$ for some \( t<2 \) $t<2$ .