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.
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