Marstrand's density theorem in the Heisenberg group

We prove that if $\mu$ is a Radon measure on the Heisenberg group ${\mathbb{H}}^n$ such that the density ${\mathrm{\Theta }}^s(\mu ,\cdot )$, computed with respect to the Kor'anyi metric $d_H$, exists and is positive and finite on a set of positive $\mu$ measure, then $s$ is an integer. The proof relies on an analysis of uniformly distributed measures on $({\mathbb{H}}^n,d_H)$. We provide a number of examples of such measures, illustrating both the similarities and the striking differences of this sub-Riemannian setting from its Euclidean counterpart.