Separated nets in Euclidean space and Jacobians of biLipschitz maps

TL;DR

This paper demonstrates that certain separated nets in the Euclidean plane cannot be transformed into the integer lattice via biLipschitz maps, using a novel construction of a continuous function that is not a Jacobian of such maps.

Contribution

It introduces a new method to distinguish separated nets from lattices by constructing a continuous function that cannot serve as a Jacobian of biLipschitz maps.

Findings

Existence of separated nets not biLipschitz equivalent to the integer lattice

Construction of a continuous function not realizable as a Jacobian of biLipschitz maps

New techniques for analyzing the geometry of nets in Euclidean space

Abstract

We show that there are separated nets in the Euclidean plane which are not biLipschitz equivalent to the integer lattice. The argument is based on the construction of a continuous function which is not the Jacobian of a biLipschitz map.