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