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

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 ≤ 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 ≥ 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.