7 problems

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…

Let $\Lambda$ be an $\operatorname{ASL}_d(\mathbb R)$-invariant point process. A cut-and-project alternative conjecture. Either $\Lambda$ is a Poisson point process, or there exist…

Let $\Gamma \subset \mathbb{R}^m \times \mathbb{R}^n$ be a lattice, let $p_2$ denote projection onto the second factor, and let $W \subset \mathbb{R}^n$ be the half-open parallelot…

Polynomial Szemerédi-type conjecture. The set $S_F$ is syndetic. This is proposed as a polynomial strengthening of the preceding corollary, motivated by polynomial extensions of Sz…

Let $G$ be a locally compact second countable group and let $\Lambda \subset G$ be a cut-and-project set. A subset of $G$ is an arithmetic progression of length $r$ if it has the f…

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

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…