## 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 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^{d}log^{d-1}T).

Original language | English |
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Pages (from-to) | 1053-1074 |

Number of pages | 22 |

Journal | Annales Scientifiques de l'Ecole Normale Superieure |

Volume | 49 |

Issue number | 5 |

DOIs | |

State | Published - 1 Sep 2016 |

Externally published | Yes |

## ASJC Scopus subject areas

- Mathematics (all)