This observation is attributed to H. Furstenberg, and appears (in the case of shift-invariant sets, i.e. Cantor sets) in his beautiful Disjointness paper (in section $3$, which you can read independently from the previous ones, although the whole paper is magnificent).
A bit more general result appears in a subsequent paper of Furstenberg named "Intersections of Cantor sets and transversality of semigroups"

I'm pretty sure such a result was known much before Furstenberg's (at-least to Erdos) but you wanted some specific references.