Intervals of spectrum at the edges of the boat–star tiling spectrum
Let $\Gamma_\bigstar$ be the tile graph of the boat–star tiling: Penrose's pentagon tiling P1, generated by the six-tile substitution of Grünbaum and Shephard (Definition 3.1 of the source), with one vertex per tile and tiles adjacent when they share an edge. Let $\Sigma_\bigstar$ be the spectrum of its graph Laplacian $\Delta = D - A$; its minimum is $0$. The source expects intervals of spectrum to be hard to establish, and its numerical approximations to the integrated density of states suggest looking near the two ends of the spectrum first, where perturbative methods may apply.
Problem. Is there a $\delta > 0$ such that $[0, \delta) \cup (\max\Sigma_\bigstar - \delta, \max\Sigma_\bigstar] \subseteq \Sigma_\bigstar$?
Source: Damanik, Embree, Fillman and Mei, Question 8.2 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
- aperiodic order, graph Laplacian, Penrose tiling, spectral edges
- 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}
}
@book{grunbaum1987,
author = {Gr{\"u}nbaum, Branko and Shephard, G. C.},
title = {Tilings and Patterns},
publisher = {W. H. Freeman and Company},
address = {New York},
year = {1987}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00022, "Intervals of spectrum at the edges of the boat–star tiling spectrum", clopen.solutions.