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arXiv · 2512.21444

The Gauss circle problem for Penrose tilings

Alan Haynes·Christopher Lutsko
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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.

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Alan Haynes, Christopher Lutsko. 2025-12-24. The Gauss circle problem for Penrose tilings. https://arxiv.org/abs/2512.21444

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