On sets with unit Hausdorff density in homogeneous groups

TL;DR

This paper proves a longstanding conjecture linking unit Hausdorff density and rectifiability for sets in homogeneous groups with smooth-box norms, advancing geometric measure theory in these spaces.

Contribution

It establishes the conjecture for subsets of homogeneous groups with smooth-box norms, showing such sets are $ ext{rectifiable}$ under the given conditions.

Findings

Confirmed the conjecture in homogeneous groups with smooth-box norms.

Demonstrated that sets with unit Hausdorff density are rectifiable in this setting.

Extended geometric measure theory to a broader class of metric spaces.

Abstract

It is a longstanding conjecture that given a subset $E$ of a metric space, if $E$ has finite Hausdorff measure in dimension $\alpha\ge 0$ and $\mathscr{H}^\alpha\llcorner E$ has unit density almost everywhere, then $E$ is an $\alpha$-rectifiable set. We prove this conjecture under the assumption that the ambient metric space is a homogeneous group with a smooth-box norm.