Interior, edge intervals and edge regularity for the Robinson triangle tiling Laplacian
Let $\Gamma_\blacktriangle$ be the tile graph of the Robinson triangle tiling: the Penrose tiling by acute and obtuse golden triangles, generated by the four-tile substitution of Definition 2.4 of the source (after Frank's primer, Fig. 13), with one vertex per triangle and triangles adjacent when they share an edge. Let $\Sigma_\blacktriangle$ be the spectrum of its graph Laplacian $\Delta = D - A$, with $\min\Sigma_\blacktriangle = 0$, and $k_\blacktriangle$ its integrated density of states, normalised per tile. $k_\blacktriangle$ jumps at $E = 2$ and at $E = 4$, each time by at least $(65 - 29\sqrt5)/20 \approx 0.0077$ (Theorem 1.5 of the source).
Problem. Answer the following for the Robinson triangle tiling. The source poses them first for the boat–star tiling.
- Does $\Sigma_\blacktriangle$ have nonempty interior? If it does, is the interior dense in $\Sigma_\blacktriangle$?
- Is there a $\delta > 0$ such that $[0, \delta) \cup (\max\Sigma_\blacktriangle - \delta, \max\Sigma_\blacktriangle] \subseteq \Sigma_\blacktriangle$?
- On a neighbourhood of $\min\Sigma_\blacktriangle$, or of $\max\Sigma_\blacktriangle$, is $k_\blacktriangle$ $\alpha$-Hölder continuous for some $\alpha \in (0, 1)$? Is it Lipschitz continuous?
Source: Damanik, Embree, Fillman and Mei, Question 8.7 in “Discontinuities of the integrated density of states for Laplacians associated with Penrose and Ammann–Beenker tilings”, Experimental Mathematics 33 (2024); arXiv:2209.01443.
Imported from arxiv.org/abs/2209.01443.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- DS, MP
- Keywords
- Cantor spectrum, graph Laplacian, integrated density of states, Penrose tiling, Robinson triangles
- Literature checked
- Oct. 1, 2026 by clemens-admin
References
@article{damanik2024,
author = {Damanik, David and Embree, Mark and Fillman, Jake and Mei, May},
title = {Discontinuities of the integrated density of states for {L}aplacians associated with {P}enrose and {A}mmann--{B}eenker tilings},
journal = {Experimental Mathematics},
volume = {33},
number = {4},
pages = {588--610},
year = {2024},
doi = {10.1080/10586458.2023.2206589},
eprint = {2209.01443},
archivePrefix = {arXiv}
}
@article{frank2008,
author = {Frank, Natalie Priebe},
title = {A primer of substitution tilings of the {E}uclidean plane},
journal = {Expositiones Mathematicae},
volume = {26},
number = {4},
pages = {295--326},
year = {2008}
}
@article{lenz2005,
author = {Lenz, Daniel and Stollmann, Peter},
title = {An ergodic theorem for {D}elone dynamical systems and existence of the integrated density of states},
journal = {Journal d'Analyse Math{\'e}matique},
volume = {97},
pages = {1--24},
year = {2005}
}
@misc{colbrook2024,
author = {Colbrook, Matthew J. and Embree, Mark and Fillman, Jake},
title = {Computing spectral size: rigorous algorithms and the limits of computation},
year = {2024},
eprint = {2407.20353},
archivePrefix = {arXiv},
note = {v2, 2026; accepted in Communications of the AMS. Section 6 and the conjectures in Section 7 concern this operator, as model T1}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00027, "Interior, edge intervals and edge regularity for the Robinson triangle tiling Laplacian", clopen.solutions.