The Heisenberg group \(H\) can be equipped with a natural (Heisenberg) left-invariant metric, given by \(d_H(p,q)=| p^{-1}\ast q|\), where \(\ast\) is the group multiplication and \(|(x,t)|=(| x|^4+t^2)^{1/4}\), \((x,t)\in H\), \(x\in{\mathbb R}^2\), \(t\in{\mathbb R}\). This metric makes \(H\) into a sub-Riemannian manifold and is not locally bi-Lipschitz equivalent to any Riemannian metric, in particular, the metric properties of \(d_H\) and the standard Euclidean metric on \(H={\mathbb R}^3\) are quite different. Comparing the two geometries makes for a very interesting research program. The authors derive their inspiration from a version of Gromov's problem calling for determining the range of possible values of the Hausdorff dimension of a subset \(S\subseteq (H,d_H)\) with a given value of the Hausdorff dimension computed with regard to the Euclidean structure on \(H\). Building on the recent advance [\textit{Z. M. Balogh, M. Rickly, F. Serra Cassano}, Publ. Mat., Barc. 47, No. 1, 237--259 (2003; Zbl 1060.28002)], the authors manage to crack the remaining particular case and thus obtain, jointly with the results from the above article, a complete answer to Gromov's problem (which is too complicated to be stated here). Proofs use a wide variety of techniques, in particular self-similar fractals, iterated affine function systems and their representations as quotients of shifts.