For s≤2 affine-invariant sets with weak specification, Hausdorff-box dimension equality is equivalent to positive finite gauge Hausdorff measure and maximal entropy measure attaining the set dimension; counterexamples exist for s≥3, and (A)⇔(B) holds for sponges.
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On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets
For s≤2 affine-invariant sets with weak specification, Hausdorff-box dimension equality is equivalent to positive finite gauge Hausdorff measure and maximal entropy measure attaining the set dimension; counterexamples exist for s≥3, and (A)⇔(B) holds for sponges.