  It is known that in $\mathbb{R}^n,n\geq 2$, a compact set which contains
$n-1$ spheres with all radii in $[1/2,1]$ 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.
