On rectifiability in Heisenberg groups without a lower density condition

We resolve a problem posed by Mattila, Serapioni and Serra Cassano concerning the role of density assumptions in the characterization of rectifiable sets of low codimension in Heisenberg groups. Specifically, we prove that the positive lower density condition is not required: rectifiability is completely determined by the geometric property of almost everywhere existence of approximate tangent subgroups. This provides a simplified and intrinsic criterion for rectifiable sets in the sub-Riemannian setting and sharpens the analogy with Federer’s classical Euclidean theorem.