Does someone know an example of a set $E$ with positive finite $s$-Hausdorff measure, Minkowski dimension $s$, and infinite $s$-dimensional upper Minkowski content ?
The $s$-dimensional upper Minkowski content is defined by \begin{equation*} \limsup\limits_{r\rightarrow 0} r^{s-d}\text{Vol}(E+B(0,r)) \end{equation*}
As you probably know, the set $\{n^{-1}:n\in\mathbb N\}\subset \mathbb R$ has Minkowski dimension $1/2$. More generally, $\{n^{-p}:n\in\mathbb N\}\subset \mathbb R$ has dimension $1/(p+1)$ because the $r$-neighborhoods are disjoint as long as $r\lesssim n^{-p}-(n+1)^{-p} \approx n^{-p-1}$. The upper Minkowski content is finite, but this can be changed, without changing the dimension, by a logarithmic factor. That is, let $$A = \{n^{-1}\log n:n\in\mathbb N\}$$ which has Minkowski dimension $1/2$ and infinite content. Key fact: the distance between $n^{-1}\log n$ and $(n+1)^{-1}\log (n+1)$ is roughly $n^{-2}\log n$, so the $r$-neighborhoods are disjoint for $n\lesssim \sqrt{\log(1/r)/r}$.
To take care of the Hausdorff requirement, take the union of $A$ with a Cantor-type set of dimension $1/2$.