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 \( R \rightarrow \infty \). 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 \( t<2 \).