Minkowski dimension behaviour of sets with positive finite Hausdorff measure.

Question

It is (rather) well known that the set \begin{equation*} E=\{k^{-1},k\in\mathbb{N}^{*}\} \end{equation*} has box-dimension $1/2$ and Hausdorff dimension $0$. However $H^{0}(E)=|E|=+\infty$.

Is it possible to find a set $E$ with Minkowski dimension $s$ for which $0<H^{s'}(E)<+\infty$ for some $s'\neq s$ ? I haven't been able to construct one, neither in $\mathbb{R}$ nor in $\mathbb{R}^{d}$. I'm enclined to think that it is possible, since geometric measure theory books (Falconer, Perrti-Mattila) heavily insist on the fact that it is very difficult to get results relating Minkowski and Hausdorff dimensions.

Note that if such a set exists then $s>s'>0$ automatically.

Answer 1

A requested example was given by Mark McClure in a comment:

How about $E\cup C$, where $E$ is your set above and $C$ is a Cantor type set of dimension $s'<1/2$?

That is, consider the union of a countable set of Minkowski dimension $1/2$ and of a self-similar Cantor set of Hausdorff (and Minkowski) dimension $s'<1/2$.

• The Minkowski dimension of the union is $1/2$.
• The Hausdorff dimension of the union is $s'<1/2$.
• The $s'$-dimensional Hausdorff measure is positive and finite on the union.