Finite Hausdorff-dimension realizations

Determine under which assumptions on an atomless probability measure \Theta a realizing metric space can be chosen to have finite Hausdorff dimension.

Background

The constructions produce realizing spaces that are topologically zero-dimensional and, in the bounded-support case, homeomorphic to the Cantor space. However, the argument provides no bound on their Hausdorff dimension. The unresolved problem is to identify conditions on the target distance law that permit a realization with finite Hausdorff dimension.

References

The spaces constructed above are zero-dimensional in the topological sense, but our argument does not give any bound on their Hausdorff dimension. It is therefore natural to ask under which assumptions on \Theta the realising space can be chosen of finite Hausdorff dimension.

— Realising atomless laws as distance distributions on metric measure spaces  (2608.28330 - Thäle et al., 28 Aug 2026) in Section 5, item (v) of the enumerated future-research questions