Suppose E is a compact set in Euclidean space which satisfies a d-dimensional topological nondegeneracy condition at all scales, and whose d-dimensional Jones beta-numbers stay larger than epsilon. Then the Hausdorff dimension of E should be larger than d. A simple case (Jones) is when d=1 and E is connected. I only know a complicated argument with quasiminimal sets, which I shall try to discuss.
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