6 problems

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…

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

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 $A$ 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…