Delone sets associated with badly approximable triangles

TL;DR

The paper constructs Delone sets from subdivision rules based on triangles with angles $\alpha,\beta,\gamma=\pi x,\pi y,\pi z$ where $x,y,z$ are badly approximable and satisfy $x+y+z=1$, aiming for rotationally invariant diffraction with minimal angular discrepancy. It proves exact classifications for the restricted case ${\mathcal{B}}_{2,1}$, obtaining two explicit solutions to $x+y+z=1$ and four to $x+y=z$, and shows via continued-fraction insertions that infinitely many explicit solutions exist in ${\mathcal{B}}_2^*$ and ${\mathcal{B}}_3$. Using a graph-directed IFS together with a threshold procedure, it constructs finite patches and proves the existence of a Delone limit set in the Chabauty--Fell topology, enabling a Delone tiling associated to optimal badly approximable angles. The approach yields stationary tilings with presumed rotationally invariant diffraction and minimal discrepancy, linking Diophantine properties of $x,y,z$ to geometric and spectral features of the resulting Delone sets. These results provide explicit constructions and open problems connecting badly approximable numbers, multiscale tilings, and diffraction theory in non-periodic order.

Abstract

We construct new Delone sets associated with badly approximable numbers which are expected to have rotationally invariant diffraction. We optimize the discrepancy of corresponding tile orientations by investigating the linear equation $x+y+z=1$ where $πx$, $πy$, $πz$ are three angles of a triangle used in the construction and $x$, $y$, $z$ are badly approximable. In particular, we show that there are exactly two solutions that have the smallest partial quotients by lexicographical ordering.

Figures (9)

• Figure 1: The pinwheel subdivision rule.
• Figure 2: The pinwheel tiling and its associated Delone set.
• Figure 3: Subdivision rule for triangles with angles $\alpha$, $\beta$, $\gamma$. The triangle on the left is scalene, and the triangle on the right is isosceles. This rule is valid for any solutions of $\alpha+\beta+\gamma=\pi$.
• Figure 4: A new tiling by optimal badly approximable triangles and its associated Delone set, constructed via the subdivision rule shown in Figure \ref{['fig:subdivision-rule-new']} with $\alpha = (2-\sqrt{3})\pi$ and ${\beta=\gamma=\frac{(\sqrt{3}-1)\pi }{2}}$.
• Figure 5: Subdivision rule for the isosceles triangle with optimal badly approximable angles $\alpha$ and $\beta=\gamma$ as in \ref{['eq:optimal1-angles']}.

Theorems & Definitions (33)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Corollary 2.5
• Remark 2.6
• Remark 2.7
• Theorem 2.8
• Theorem 2.9