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][1] (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][2]"

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


  [1]: http://www.math.tau.ac.il/~barakw/seminar/furstenberg_disjointness.pdf
  [2]: https://books.google.co.il/books?hl=en&lr=&id=ZXh9BgAAQBAJ&oi=fnd&pg=PA41&dq=intersection%20of%20cantor%20sets&ots=dG_CShgKVz&sig=Jb70qXl_SLACRQv12j3IxyOe7X4&redir_esc=y#v=onepage&q=intersection%20of%20cantor%20sets&f=false