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 discuss an asymptotic count for S_{\alpha}(\epsilon,R), the number of integer lattice points lying in such a sector, and moreover present results concerning the variance of such lattice points across sectors.