## Abstract

Considering the space of closed subsets ofRd, endowed with the Chabauty-Fell topology, and the affine action of SLd(R)Rd, we prove that the only minimal subsystems are the fixed points and {Rd}. As a consequence we resolve a question of Gowers concerning the existence of certain Danzer sets: there is no set Y Rd such that for every convex set C Rd of volume one, the cardinality of C n Y is bounded above and below by non-zero constants independent of C. We also provide a short independent proof of this fact and deduce a quantitative consequence: for every e-net N for convex sets in [0, 1]d there is a convex set of volume e containing at least (log log(1/e)) points of N.

Original language | English |
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Pages (from-to) | 6584-6598 |

Number of pages | 15 |

Journal | International Mathematics Research Notices |

Volume | 2017 |

Issue number | 21 |

DOIs | |

State | Published - 1 Nov 2017 |

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