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The Gauss circle problem for Penrose tilings

Alan Haynes, Christopher Lutsko

TL;DR

This work resolves a Gauss-circle-type counting problem for the vertex set of unit-length rhombic Penrose tilings by representing Penrose vertices as cut-and-project sets in a 4-dimensional lattice and applying Fourier-analytic Poisson summation with carefully smoothed indicators. The main result shows that, for $R\ge 2$, the number of Penrose-tiling vertices in the disk $B_R$ satisfies $\#(V\cap B_R)=\pi C_P R^2 + O(R^{2/3}(\log R)^{2/3})$, where $C_P=\frac{\phi}{5}\sqrt{10+2\sqrt{5}}\approx1.231$ and $\phi=(1+\sqrt{5})/2$, with the density constant $\mathrm{dens}(\mathcal{L})=\frac{4}{25\sqrt{5}}$. The analysis combines the cut-and-project framework (Minkowski embedding of an algebraic lattice, pentagonal windows) with a delicate, anisotropic smoothing and a careful treatment of Fourier coefficients, especially in directions perpendicular to tiling edges. By bounding the dual-lattice contribution in three regimes and optimizing smoothing parameters, the authors achieve a uniform error bound valid for all non-singular window choices, enabling limits that yield the Penrose tiling result and connecting to the classical Gauss circle problem. The explicit constants arise from window areas and lattice density, reinforcing the link between quasicrystal geometry and lattice-point discrepancy theory, and suggesting extensions to other aperiodic tilings such as Ammann-Beenker 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 $R\ge 2$, \[\#(V\cap B_R)=πC_P R^2 + O(R^{2/3}(\log R)^{2/3}),\] where $C_P\approx 1.231$ is a constant.

The Gauss circle problem for Penrose tilings

TL;DR

This work resolves a Gauss-circle-type counting problem for the vertex set of unit-length rhombic Penrose tilings by representing Penrose vertices as cut-and-project sets in a 4-dimensional lattice and applying Fourier-analytic Poisson summation with carefully smoothed indicators. The main result shows that, for , the number of Penrose-tiling vertices in the disk satisfies , where and , with the density constant . The analysis combines the cut-and-project framework (Minkowski embedding of an algebraic lattice, pentagonal windows) with a delicate, anisotropic smoothing and a careful treatment of Fourier coefficients, especially in directions perpendicular to tiling edges. By bounding the dual-lattice contribution in three regimes and optimizing smoothing parameters, the authors achieve a uniform error bound valid for all non-singular window choices, enabling limits that yield the Penrose tiling result and connecting to the classical Gauss circle problem. The explicit constants arise from window areas and lattice density, reinforcing the link between quasicrystal geometry and lattice-point discrepancy theory, and suggesting extensions to other aperiodic tilings such as Ammann-Beenker tilings.

Abstract

Let denote the closed Euclidean ball of radius in the plane. In this paper we prove that, if is the set of vertices of any unit length rhombic Penrose tiling then, for , where is a constant.
Paper Structure (5 sections, 8 theorems, 67 equations, 2 figures)

This paper contains 5 sections, 8 theorems, 67 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Vertices of Penrose tilings via cut and project sets
  3. Results from Fourier analysis
  4. Setup for counting estimates
  5. Analysis of error terms: Proofs of Theorems \ref{['thm.Pen']} and \ref{['thm.PenCPVersion']}

Key Result

Theorem 1

Let $V$ be the set of vertices of any unit length RPT. Then, for $R\ge 2$,

Figures (2)

  • Figure 1: Tiles used to construct rhombic Penrose tilings.
  • Figure 2: A patch of an RPT constructed using the cut and project method detailed in Section \ref{['sec.ModelSets']}. The 4 colors of vertices are translates of the cut and project sets produced using the different windows.

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • Proposition 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8