Spectral gaps of 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 $\Delta = D - A$. The eigenvalues of finite patches, up to level 15 of the substitution, show several gaps that appear to persist as the level grows, among them gaps a little below and a little above $E = 3$; the source's authors suspect there are infinitely many (Fig. 27 of the source).
Problem. Show that $\Sigma_\blacktriangle$ has a nontrivial spectral gap, that is, that it is not an interval. Does it have infinitely many gaps?
Source: Damanik, Embree, Fillman and Mei, Question 8.8 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, Penrose tiling, Robinson triangles, spectral gap
- 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}
}
@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-00028, "Spectral gaps of the Robinson triangle tiling Laplacian", clopen.solutions.