2016/01/01 by Yaar Solomon, Barak Weiss · 1 citation
Mathematics · Computer Science · #Markov Chains and Monte Carlo Methods #Topological and Geometric Data Analysis #Computational Geometry and Mesh Generation
paper · doi:10.24033/asens.2303
openalex publication_date 2016/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A set Y⊆ℝd 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(Td). 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 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(Tdlog T), improving the previous bound of O(Tdlogd-1 T).