Let E be a compact set in Euclidean space which satisfies a d-dimensional topological nondegeneracy condition at all scales, and whose d-dimensional Peter Jones numbers stay larger than epsilon. Then the Hausdorff dimension of E should be larger than d. A simple case (Bishop-Jones) is when d=1 and E is connected.
The only proof I know so far uses the uniform rectifiability of Almgren quasiminimal (restricted) sets.