Abstract. A set Y ⊆ R d that intersects every convex set of volume 1 is called a Danzer set. It is not known whether there are Danzer sets in R d with growth rate O(T d ). We prove that natural candidates, such as discrete sets that arise from substitutions and from cut-and-project constructions, are not Danzer sets. For cut and project sets our proof relies on the dynamics of homogeneous flows. We consider a weakening of the Danzer problem, the existence of a uniformly discrete dense forests, and we use homogeneous dynamics (in particular Ratner's theorems on unipotent flows) to construct such sets. We also prove an equivalence between the above problem and a well-known combinatorial problem, and deduce the existence of Danzer sets with growth rate O(T d log T ), improving the previous bound of O(T d log d−1 T ).

We show that any primitive substitution tiling of R 2 creates a separated net which is biLipschitz to Z 2 . Then we show that if H is a primitive Pisot substitution in R d , for every separated net Y , that corresponds to some tiling τ ∈ X H , there exists a bijection Φ between Y and the integer lattice such that sup y∈Y Φ(y) − y < ∞. As a corollary, we get that we have such a Φ for any separated net that corresponds to a Penrose Tiling. The proofs rely on results of Laczkovich, and Burago and Kleiner.

We study separated nets Y that come from primitive substitution tilings of the Euclidean space R d . We show that the question whether Y is a bounded displacement of Z d or not can be reduced, in most cases, to a simple question on the eigenvalues and eigenspaces of the substitution matrix.

We introduce a new general framework for constructing tilings of Euclidean space, which we call multiscale substitution tilings. These tilings are generated by substitution schemes on a finite set of prototiles, in which multiple distinct scaling constants are allowed. This is in contrast to the standard case of the well‐studied substitution tilings which includes examples such as the Penrose and the pinwheel tilings. Under an additional irrationality assumption on the scaling constants, our construction defines a new class of tilings and tiling spaces, which are intrinsically different from those that arise in the standard setup. We study various structural, geometric, statistical, and dynamical aspects of these new objects and establish a wide variety of properties. Among our main results are explicit density formulas and the unique ergodicity of the associated tiling dynamical systems.

Dense forests are discrete subsets of Euclidean space which are uniformly close to all sufficiently long line segments. The degree of density of a dense forest is measured by its visibility function. We show that cut‐and‐project quasicrystals are never dense forests, but their finite unions could be uniformly discrete dense forests. On the other hand, we show that finite unions of lattices typically are dense forests, and give a bound on their visibility function, which is close to optimal. We also construct an explicit finite union of lattices which is a uniformly discrete dense forest with an explicit bound on its visibility.