Cubes, side lengths and centres

It is known that in ${\mathbb{R}}^n,n\ge 2$, a compact set which contains $n-1$ spheres with all radii in $[1\mathrm{/}2,1]$ or with all possible centres in $[0,1{]}^n$ has full Hausdorff dimension. In fact the later set has positive Lebesgue measure. In this paper we consider a similar problem with sphere replacing by fractal cubes. The radii set and the centre set are also considered to be fractal sets. In addition we discuss the exceptional set in the setting of general largeness. In the end, an Furstenberg type example is discussed which can be somehow considered as the Furstenberg $\times 2$, $\times 3$ set conjecture (now theorem) in the setting of cubes/circles sets considered here.