The Gauss circle problem for Penrose tilings

Let $B_R$ denote the closed Euclidean ball of radius $R$ in the plane. In this paper we prove that, if $V$ is the set of vertices of any unit length rhombic Penrose tiling then, for $R\ge 2$, $$\mathrm{\#}(V\cap B_R)=\pi C_P{R}^2+O(R^{2\mathrm{/}3}(\log R{)}^{2\mathrm{/}3}),$$ where $C_P\approx 1.231$ is a constant.