On the number of lattice points in thin sectors

Abstract

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<2 $.