The Gauss circle problem for Penrose tilings

TL;DR

This paper extends the Gauss circle problem to Penrose tilings, showing that the number of vertices within a large radius approximates a scaled area with a specific error term.

Contribution

It provides the first quantitative estimate of lattice point counts in Penrose tilings, generalizing classical circle problem results to a non-periodic tiling.

Findings

Number of vertices in Penrose tilings within radius R approximates πC_P R^2

Error term in vertex count is bounded by R^{2/3} (log R)^{2/3}

Constant C_P is approximately 1.231

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.