AbstractOn the circle of radius R centred at the origin, consider a “thin” sector about the fixed line $$y = \alpha x$$ y = α x with edges given by the lines $$y = (\alpha \pm \epsilon ) x$$ y = ( α ± ϵ ) x , where $$\epsilon = \epsilon _R \rightarrow 0$$ ϵ = ϵ R → 0 as $$ R \rightarrow \infty $$ R → ∞ . We establish an asymptotic count for $$S_{\alpha }(\epsilon ,R)$$ S α ( ϵ , R ) , the number of integer lattice points lying in such a sector. Our results depend both on the decay rate of $$\epsilon $$ ϵ and on the rationality/irrationality type of $$\alpha $$ α . In particular, we demonstrate that if $$\alpha $$ α is Diophantine, then $$S_{\alpha }(\epsilon ,R)$$ S α ( ϵ , R ) is asymptotic to the area of the sector, so long as $$\epsilon R^{t} \rightarrow \infty $$ ϵ R t → ∞ for some $$ t<2 $$ t < 2 .