Equivalence relations on separated nets arising from linear toral flows

Abstract

In 1998, Burago-Kleiner and McMullen independently proved the existence of separated nets in ${\mathbb{R}}^d$ 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 ${\mathbb{R}}^d$-actions. We analyze the separated nets which arise via these constructions, focusing particularly on nets arising from linear ${\mathbb{R}}^d$-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.

33 pages, a gap in the proof of Corollary 3.5 was corrected and a few other minor modifications were made

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