Equivalence relations on separated nets arising from linear toral flows

TL;DR

This paper investigates the equivalence relations of separated nets from linear toral flows, showing that generically they are bi-Lipschitz or bounded displacement equivalent to lattices, but some are not BD equivalent.

Contribution

It provides a detailed analysis of separated nets from linear toral flows, establishing conditions for BL and BD equivalences and identifying cases where nets are not BD to lattices.

Findings

Generically, these nets are bi-Lipschitz equivalent to lattices.

Some nets are generically bounded displacement equivalent to lattices.

Existence of nets not BD equivalent to any lattice.

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.