Dimension results for graphs on compact Heisenberg subsets

Investigate and obtain dimension results for graphs of functions defined on compact subsets of the first Heisenberg group, including the lifted Sierpiński gasket, with respect to the Heisenberg and Euclidean metrics.

Background

The paper constructs an f-FIF on a non-compact subset of the first Heisenberg group whose graph has strictly greater Hausdorff dimension under the Heisenberg metric than under the Euclidean metric. The authors note that they have not established an analogous counterexample for compact subsets of the Heisenberg group.

The unresolved issue arises because the covering techniques required for compact domains differ substantially from those used for non-compact domains. The lifted Sierpiński gasket is highlighted as a compact domain for which corresponding graph-dimension results remain to be developed.