On sets with unit Hausdorff density in homogeneous groups

Abstract

Abstract It is a longstanding conjecture that given a subset E of a metric space, if E has unit $\mathscr {H}^{\alpha }\llcorner E$ -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.