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

TL;DR

This paper investigates when the Hausdorff and box dimensions of certain affine-invariant sets on the torus coincide, revealing new phenomena in higher dimensions and establishing equivalences under specific conditions.

Contribution

It proves the equivalence of dimension and measure conditions for affine-invariant sets with two eigenvalues and explores new behaviors when there are three or more eigenvalues.

Findings

For s ≤ 2, the equivalence of dimensions and measure positivity is established.

Examples show that for s ≥ 3, the dimensions may not coincide even if the measure dimension condition holds.

A probabilistic approach confirms the equivalence for Bedford-McMullen sponges.

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 (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.