On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets

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 $s\le 2$, 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 $s\ge 3$, we find some examples in which (A) does not hold but (C) holds, which is a new phenomenon not appearing in the planar cases. Through a different probabilistic approach, we establish the equivalence of (A) and (B) for Bedford-McMullen sponges.

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