A comparison of Euclidean and Heisenberg Hausdorff measures

We prove some geometric properties of sets in the first Heisenberg group whose Heisenberg Hausdorff dimension is the minimal or maximal possible in relation to their Euclidean one and the corresponding Hausdorff measures are positive and finite. In the first case we show that these sets must be in a sense horizontal and in the second case vertical. We show the sharpness of our results with some examples.