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On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets
Published online by Cambridge University Press: 08 September 2025
Abstract
Let 
$ K $ be a compact subset of the d-torus invariant under an expanding diagonal endomorphism with s distinct eigenvalues. Suppose the symbolic coding of K satisfies weak specification. When 
$ s \leq 2 $, we prove that the following three statements are equivalent: (A) the Hausdorff and box dimensions of 
$ K $ coincide; (B) with respect to some gauge function, the Hausdorff measure of 
$ K $ is positive and finite; (C) the Hausdorff dimension of the measure of maximal entropy on 
$ K $ attains the Hausdorff dimension of 
$ K $. When 
$ s \geq 3 $, we find some examples in which statement (A) does not hold but statement (C) holds, which is a new phenomenon not appearing in the planar cases. Through a different probabilistic approach, we establish the equivalence of statements (A) and (B) for Bedford–McMullen sponges.
MSC classification
Primary:
28A80: Fractals
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- Original Article
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- © The Author(s), 2025. Published by Cambridge University Press
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