19 problems

Period Conjecture. For each integer $j=1,\dots,d$, there is a positive constant $c_j(r,R)$ such that, if for some $t_0(X)$

Absolutely continuous diffraction conjecture. The absolutely continuous component of the diffraction measure is nonzero:

Let $D\subset\mathbb{R}^d$ be a Delone set. A Delone set is aperiodic if it has no nonzero translational period, linearly repetitive if every finite pattern occurring in it reappea…

Let $G(V,E)$ be a connected infinite graph with bounded vertex degrees. Assume that $G$ is amenable and is an abstract quasicrystal graph: for every radius $r$ and every $r$-patter…

Let $d\geq 2$ and let $\Lambda=\Lambda(\mathcal{W},\mathcal{L})\subset\mathbb{R}^d$ be an irreducible cut-and-project set. Let $\mathcal{A}\subset\mathbb{R}^d$ be a van Hove sequen…

0-1 law conjecture. On sufficiently regular rhombus tilings, including Penrose and multigrid dual tilings, the probability of percolation is either $0$ or $1$.

Singularity conjecture. The continuous component $\widehat{\gamma}^{\ }_{\textsf{cont}}$ must be singular, namely singular continuous rather than absolutely continuous.

Let $\mathcal T$ be a tiling with finite local complexity. For each patch class $\mathcal P$ of $\mathcal T$, let $\nu(\mathcal P)$ denote its frequency, and let…

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

Let $X\subset\mathbb{R}^3$ be a Delone set with packing radius $r$, covering radius $R$, regularity radius $\widehat{R}$, and $N(2R)=1$. A local $5$-fold symmetry means that the…

A two-dimensional Rauzy tiling is a non-periodic uniformly repetitive tiling whose corona limit is defined by the asymptotic shape of its coronas. Octagon conjecture. The corona li…

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

The paper constructs an aperiodic point space from higher-dimensional crystal structures, specifically relating an icosahedral quasicrystal to quasicrystals derived from the $E_8$…

Let $(G\times H,\mathcal{L})$ be a cut and project scheme, let $W\subset H$ be a compact set, and let \text{\Large curlywedge}(W) denote the associated model set. Write…

For a parameter $\lambda\in\mathbb C$, call the triangle with vertices $0$, $1$, and $\lambda$ the fundamental triangle. Let $R_\lambda$ be the associated closure. McNulty's fundam…

Let $R_\lambda$ be the set associated with the parameter $\lambda$ in the paper. The authors previously conjectured that $R_\lambda$ is convex for every real $\lambda$ strictly bet…

Let the Ammann-Beenker tilings be the tilings formed by the tiles shown in Fig. … tiles is one, so that exactly all these tilings can be formed. This would sharpen the proved unifo…

Let $A$ be a finite alphabet, and let a tiling in ${\mathbb R}^d$ have a suitably defined notion of linear repetitivity. Its transversal is the tiling space consisting of tilings l…

A particle system consists of finitely many types of particles placed on a discrete point set, with stochastic deviations from a system having a strongly ordered ground state. Its…