  In Carnot groups of step 2 we consider sets having maximal or minimal
possible homogeneous Hausdorff dimension compared to their Euclidean one: in
the first case we prove that they must be in a sense vertical, that is a large
part of these sets can lie off horizontal planes through their points, and in
the latter case close to horizontal. The examples that had been used to prove
the sharpness of the sub-Riemannian versus Euclidean dimension comparison
intuitively satisfied these geometric properties, which we here quantify and
prove to be necessary. We also show the sharpness of our statements with some
examples. This paper generalizes the analogue results proved by Mattila and the
author in the first Heisenberg group.
