45 problems

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 $\lambda$.

Last's conjecture. In general, $\dim_B^+\Sigma_\lambda$ does not bound $\tilde\alpha^\pm_u(\lambda)$ from above.

The rank-of-periods conjecture. For each $j$ with $1\leq j\leq n$, there is a positive constant $c_j=c_j(n,r,R)$ such that, if

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.

Fractal-window conjecture. All square-triangle tilings with 12-fold symmetry obtained by projection require a fractally shaped window. In a certain sense, the window in the example…

Let the paperfolding Hamiltonian be the Hamiltonian generated by the paperfolding substitution sequence, with coupling parameter $\lambda$. Paperfolding spectral conjecture. The pa…

Let $\beta(\alpha)$ denote the exponent governing the decay of the return probability, and let $D_1$ and $D_0$ denote, respectively, the information dimension of the relevant spect…

Consider the four-dimensional finite-range lattice-gas model with blank particles and non-negative interaction energies, whose ground states include the all-blank configuration and…

Consider the four-dimensional finite-range lattice-gas model whose ground-state configurations include standard configurations, constant in the first two directions and correspondi…

Typical diffraction-density conjecture. For any $M,r>0$ there exists $\epsilon>0$ such that $\Lambda_{u,\epsilon}$ is $r$-dense in ${\mathcal L}^*\cap B(0,M)$.

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 $F$ be the limiting gap distribution function for the cases considered, and let $F'(s)$ denote its derivative. Conjecture. In the cases considered, $F'(s)$ exists, is continuou…

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

The $3,5,3 honeycomb is a hyperbolic honeycomb in$operatorname{mathbb{H}}^3…

Let $\alpha$ be a relatively uniformly discrete multiset in $\mathbb R^n$, meaning that it is a finite sum of uniformly discrete multisets. The associated construction gives a bran…

Let $\alpha$ be a uniformly discrete Bohr almost periodic multiset in $\mathbb R^n$. Its orbit closure has a toral compactification, and under the stated finite-rank hypothesis thi…

Sturmian Dry Ten Martini conjecture. For every irrational $\alpha\in[0,1]$ and every coupling $V\neq 0$, the Sturmian Dry Ten Martini Problem is true; equivalently,

Gapless-topology inheritance conjecture. Projected topological branes possibly inherit gapless topology from the parent gapless phase.

Gap-opening and gap-labelling conjecture. All possible gaps of $\sigma_\lambda$ are open, and their labels are dyadic numbers or dyadic numbers divided by $3$ in $(0,1)$.

Lagarias–Pleasants minimal complexity conjecture. Every non-periodic pattern satisfies

Let ${\boldsymbol{R}}:\mathbb{R}^\mathbbl{n}\to\mathbb{R}^\mathbbl{m}$ be the cut-and-project map, and let $\sigma_R$ be the function associated with $\sigma$ by the definition in…

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