20 problems

Take the $\{3,5,3\}$ honeycomb in $\operatorname{\mathbb{H}}^3$ and cut it along a horosphere. Consider the union of all icosahedra interior to, or intersecting, the horosphere; th…

The patch-counting lower-bound conjecture. For every dimension $n\geq 1$ and Delone constants $(r,R)$, there is a positive constant $c=c(n,r,R)$ such that every aperiodic Delone se…

The linear repetitivity conjecture. Every aperiodic linearly repetitive Delone set is densely repetitive.

Bellissard's gap-labelling conjecture.

For an irrational $\alpha\in(0,1)$, let $\sigma_{\lambda,\alpha}$ denote the corresponding Sturmian spectrum and let $\dim_H$ denote Hausdorff dimension. Small-coupling continuity…

Let $H_{\lambda,\alpha,\omega}$ be the Sturmian Hamiltonian with coupling $\lambda>0$, irrational frequency $\alpha\in(0,1)$, and phase $\omega\in S^1$, and let…

Let the $3,5,3 honeycomb be a tessellation of$operatorname{mathbb{H}}^3…

Let $\sigma$ be a Pisot irreducible substitution, meaning that the characteristic polynomial of its incidence matrix is the minimal polynomial of a Pisot number. Let…

Periodic approximability conjecture. The transversal of $\mathcal{D}$ is periodically approximable if $\mathcal{D}$ is defined via a cut-and-project scheme.

Coincidence Rank Conjecture. The coincidence rank of the tiling must divide the algebraic norm of $\lambda$.

Homological Pisot Conjecture. The tiling has pure point spectrum if its first rational Čech cohomology group has rank equal to the algebraic degree of $\lambda$.

Pisot Substitution Conjecture. The tiling has pure point spectrum if $M$ has an irreducible characteristic polynomial; equivalently, if the algebraic degree of $\lambda$ equals the…

Cantorval spectrum conjecture. There exist values of $\lambda_1$ and $\lambda_2$ such that the spectrum $\sigma(H)$ is a Cantorval; that is, the spectrum is the closure of its inte…

Pattern Sturmian spectral conjecture. Theorem 1 holds if $\bm{x}$ is a pattern Sturmian sequence: the spectrum $\sigma(H)$ is a zero Lebesgue measure Cantor set and all spectral me…

Let $d>1$, and let $\mathcal{C}_d'$ be the collection of convex sets in $\mathbb{R}^d$ with inradii at least $1/2$. A cut-and-project set is called cubical or canonical as in the p…

Let $\Lambda$ be the quasicrystal with transversal $\Omega_{trans}$, and let $\mathcal{A}_{\theta=0}$ be its twisted groupoid $C^*$-algebra for zero magnetic field. The trace induc…

Let $d706>0$ and $d6fc\in(0,1)\setminus\mathbb{Q}$, and let $G(\alpha)$ denote the Gauss map applied to $d6fc$. Write $d6a9_{\lambda,\alpha}$ and $d6a9_{\lambda,G(\alpha)}$ for the…

Let $d706>0$ and let $d6fc\in(0,1)\setminus\mathbb{Q}$ be an irrational frequency. Denote by $d6a9_{\lambda,\alpha}$ the phase-independent spectrum of the corresponding Sturmian Sc…

Let $E_\parallel$ be a $d$-dimensional linear subspace of ${\mathbb R}^n$ with irrational orientation, and let $\mathcal L(E_\parallel)$ be the associated cut-and-project Delone se…

Let $J_0,J_1>0$, and let $\{\sigma_k\}$ be the sequence of spectra whose Hausdorff limit is studied in the paper. Let $B_\infty(J_0,J_1)$ denote the limiting spectrum when this lim…