Dense forests and Danzer sets

2014/06/15 by Yaar Solomon, Solomon, Yaar, Barak Weiss +1 · 1 citation
Computer Science · Mathematics · #Computational Geometry (cs.CG) #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #Metric Geometry (math.MG) #cs.CG #math.DS #math.MG

paper · pdf · doi:10.48550/arxiv.1406.3807

arxiv created 2014/07/10 · arxiv updated 2014/07/14

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 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^dlog T), improving the previous bound of O(T^dlog^d-1 T).

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