## Abstract

In 1998, Burago-Kleiner and McMullen independently proved the existence of separated nets in ℝ^{d} which are not bi-Lipschitz equivalent (BL) to a lattice. A finer equivalence relation than BL is bounded displacement (BD). Separated nets arise naturally as return times to a section for minimal ℝ^{d}-actions. We analyze the separated nets which arise via these constructions, focusing particularly on nets arising from linear ℝ^{d}-actions on tori. We show that generically these nets are BL to a lattice, and for some choices of dimensions and sections, they are generically BD to a lattice. We also show the existence of such nets which are not BD to a lattice.

Original language | English |
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Pages (from-to) | 1203-1228 |

Number of pages | 26 |

Journal | Proceedings of the London Mathematical Society |

Volume | 109 |

Issue number | 5 |

DOIs | |

State | Published - 9 May 2013 |