Asymptotic estimates of large gaps between directions in certain planar quasicrystals

TL;DR

This work analyzes the asymptotic tails of the limiting gap distribution for directions to points in planar cut-and-project sets (planar quasicrystals). Building on the existence and Halvers of SL$_d(\mathbb{R})$-invariant limit processes for directions, the authors derive precise tail estimates: $G(s)\sim a_{\mathcal{P}}/s$ and $F(s)\sim a_{\mathcal{P}}/s^2$ as $s\to\infty$, with refined error bounds under convex windows. The leading coefficient $a_{\mathcal{P}}$ is given by an explicit, computable formula involving ideal data from real quadratic fields, the window, and the Dedekind zeta function; in special cases it reduces to a Mahler-type product $\frac{\operatorname{Area}(\mathcal{W})\operatorname{Area}(\mathcal{W}^*)}{\Delta_K^2\zeta_K(2)}$, with $\mathcal{W}^*$ the polar body of $\mathcal{W}$. The paper applies the framework to Ammann-Beenker, Gähler's shield, and Tübingen triangle tilings, delivering explicit expressions and numerical verifications, and discusses implications for related dynamical systems and transport problems in quasi-crystal geometries. Overall, it connects tail gap statistics to deep number-theoretic and homogeneous dynamics structures, enabling precise, computable predictions for large-gap behavior in planar quasicrystals.

Abstract

For quasicrystals of cut-and-project type in $\mathbb{R}^d$, it was proved by Marklof and Strömbergsson that the limit local statistical properties of the directions to the points in the set are described by certain $\operatorname{SL}_d(\mathbb{R})$-invariant point processes. In the present paper we make a detailed study of the tail asymptotics of the limiting gap statistics of the directions, for certain specific classes of planar quasicrystals.

Figures (2)

• Figure 1: Patches of an Ammann-Beenker tiling with vertex set ${\mathcal{P}}_{\operatorname{AB},(0.8,0.1)}$ (top left panel), a Gähler's shield tiling with vertex set ${\mathcal{P}}_{\operatorname{Gh},(1.7,0.6)}$ (top right panel), and Tübingen triangle tilings with vertex sets ${\mathcal{P}}_{\operatorname{\space T \space T},(0.001,0)}$ (bottom left panel) and ${\mathcal{P}}_{\operatorname{\space T \space T},(1.3,0.4)}$ (bottom right panel). The notation is explained in Section \ref{['examplessec']}. In each patch, the origin is marked by a small black disc.
• Figure 2: Experimental graphs of the limiting density of normalized gaps, $-F'(s)$, for the vertex sets of Ammann-Beenker tilings ${\mathcal{P}}_{\operatorname{AB},{\text{\boldmath$w$}}}$ with (I) ${\text{\boldmath$w$}}=\mathbf{0}$ and (II) ${\text{\boldmath$w$}}=(0.9,0.3)$; Gähler's shield tilings ${\mathcal{P}}_{\operatorname{Gh},{\text{\boldmath$w$}}}$ with (III) ${\text{\boldmath$w$}}=(1.7,0.6)$ and (IV) ${\text{\boldmath$w$}}=(0.1,0)$; and Tübingen triangle tilings ${\mathcal{P}}_{\operatorname{\space T \space T},{\text{\boldmath$w$}}}$ with (V) ${\text{\boldmath$w$}}=(0.5,0.4)$ and (VI) ${\text{\boldmath$w$}}=(1.3,0.4)$ (notation as in Section \ref{['examplessec']}). These graphs were obtained from numerical computations described in Section \ref{['numcompSUBSEC']}.

Theorems & Definitions (74)

• Theorem 1
• Remark 1.7
• Remark 1.9
• Theorem 2
• Remark 1.15
• Theorem 3
• Remark 1.18
• Remark 1.19
• Remark 1.20
• Remark 1.22