Dense forests and Danzer sets

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 Rd 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 forest, 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^dlog^d-1T).