Dense forests and Danzer sets

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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 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(Td log T), improving the previous bound of O(Tdlogd-1T).

Original languageEnglish
Pages (from-to)1053-1074
Number of pages22
JournalAnnales Scientifiques de l'Ecole Normale Superieure
Issue number5
StatePublished - 1 Sep 2016

All Science Journal Classification (ASJC) codes

  • Mathematics(all)


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