Some remarks on densities in the Heisenberg group

TL;DR

This paper investigates the behavior of densities on surfaces within the Heisenberg group, revealing differences between upper and spherical Federer densities and identifying rectifiable sets with densities less than one.

Contribution

It demonstrates that upper and spherical Federer densities can differ on all two-dimensional surfaces in the Heisenberg group, introducing a new class of rectifiable sets with densities below one.

Findings

Upper and spherical Federer densities may differ on surfaces in the Heisenberg group.

Existence of intrinsic rectifiable sets with upper density less than one.

Provides insight into density behavior in sub-Riemannian geometry.

Abstract

We observe that upper densities and spherical Federer densities may differ on all two dimensional surfaces of the sub-Riemannian Heisenberg group. This provides an entire class of intrinsic rectifiable sets having upper density strictly less than one.