Marstrand's density theorem in the Heisenberg group

We prove that if μ is a Radon measure on the Heisenberg group Hⁿ such that the density θ ^sμ), computed with respect to the Korányi metric d_H, exists and is positive and finite on a set of positive μ measure, then s is an integer. The proof relies on an analysis of uniformly distributed measures on Hⁿ, 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.