Geodesic
On sets with unit Hausdorff density in homogeneous groups
Forum of Mathematics, Sigma, Tome 11 (2023)

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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. 
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     author = {Antoine Julia and Andrea Merlo},
     title = {On sets with unit {Hausdorff} density in homogeneous groups},
     journal = {Forum of Mathematics, Sigma},
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     year = {2023},
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Antoine Julia; Andrea Merlo. On sets with unit Hausdorff density in homogeneous groups. Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.31

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