Equivalence Relations on Separated Nets Arising from Linear Toral Flows

In 1998, Burago-Kleiner and McMullen independently proved the existence of separated nets in R which are not bi-Lipschitz equivalent (BL) to a lattice. A finer equivalence relation than BL is bounded displacement after dilation (BDD). Separated nets arise naturally as return times to a section for minimal R-actions. We analyze the separated nets which arise via these constructions, focusing particularly on nets arising from linear R-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 BDD to a lattice. We also show the existence of such nets which are not BDD to a lattice.