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 $\epsilon ={\epsilon }_R\to 0$ as $R\to \mathrm{\infty }$. 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 $\epsilon R^t\to \mathrm{\infty }$ for some $t<2$.