Density not realizable as the Jacobian determinant of a bilipschitz map

TL;DR

This paper proves that certain density functions cannot be realized as the Jacobian determinant of bilipschitz maps, clarifying a key aspect of geometric measure theory and the structure of separated nets in the plane.

Contribution

It provides a complete proof that McMullen's density construction is not realizable as a Jacobian determinant, filling a gap in the literature.

Findings

McMullen's density is not realizable as a Jacobian determinant

Full proof of nonrealizability of McMullen's construction

Clarification of the relationship between separated nets and bilipschitz maps

Abstract

Are every two separated nets in the plane bilipschitz equivalent? In the late 1990s, Burago and Kleiner and, independently, McMullen resolved this beautiful question negatively. Both solutions are based on a construction of a density function that is not realizable as the Jacobian determinant of a bilipschitz map. McMullen's construction is simpler than the Burago-Kleiner one, and we provide a full proof of its nonrealizability, which has not been available in the literature.