On sets with unit Hausdorff density in homogeneous groups

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

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