Marstrand’s density theorem in the Heisenberg group

nina ayu rachmawaty;

We prove that if μ is a Radon measure on the Heisenberg group Hn such that the density Θs(μ, ·), computed with respect to the Kor´anyi metric dH, 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, dH). 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.