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Let d be an integer, and let E be a nonempty closed subset of Rn. Assume that E is locally uniformly non flat, in the sense that for x ∈ E and r > 0 small, E∩B(x, r) never stays ε0r-close to an affine d-plane. Also suppose that E satisfies locally uniformly some appropriate d-dimensional topological nondegeneracy condition, like Semmes’ Condition B. Then the Hausdorff dimension of E is strictly larger than d. We see this as an application of uniform rectifiability results on Almgren quasiminimal (restricted) sets

Hausdorff dimension of uniformly non flat sets with topology