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=αx with edges given by the lines y=(α±ϵ)x, where ϵ=ϵ_R→0 as R→∞. 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→∞ for some t<2.