The Gauss circle problem for Penrose tilings

Abstract

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 $R\ge 2$, \[\#(V\cap B_R)=\pi C_P R^2 + O(R^{2/3}(\log R)^{2/3}),\] where $C_P\approx 1.231$ is a constant.