Triggered by the recent question [How can we not know the measure of the Sierpiński triangle?][1] I would like to ask:

Let $s>1$ and $s$ not be an integer. How to construct a set $A$ with $\mathfrak{H}^s(A) = 1$, i.e. an $s$-dimensional set with $s$-dimensional Hausdorff measure $1$? Is there a set which in some sense "as simple as possible"?

For $0<s<1$ this is not so difficult, since covering with disjoint intervals makes the respective terms as large as possible, but I do not see how a similar argument can be made in higher dimensions.

Phrased differently: What should "the unit cube in $d$-dimensions" be for non-integer $d>1$. [1]: How can we not know the $s$-measure of the Sierpiński triangle?

Triggered by the recent question How can we not know the measure of the Sierpiński triangle? I would like to ask:

Let $s>1$ and $s$ not be an integer. How to construct a set $A$ with $\mathfrak{H}^s(A) = 1$, i.e. an $s$-dimensional set with $s$-dimensional Hausdorff measure $1$? Is there a set which in some sense "as simple as possible"?

For $0<s<1$ this is not so difficult, since covering with disjoint intervals makes the respective terms as large as possible, but I do not see how a similar argument can be made in higher dimensions.

Phrased differently: What should "the unit cube in $d$-dimensions" be for non-integer $d>1$.