A proof of Furstenberg's conjecture on the intersections of × p- and × q-invariant sets

We prove the following conjecture of Furstenberg (1969): if A,B⊂ [0,1] are closed and invariant under × p\ mod\ 1 and ×q\ mod\ 1, respectively, and if log\ p/log\ q≠ Q, then for all real numbers u and v, dimH(uA+v) ∩ B ≤ max\0,dimH A + dimH B -1\.. We obtain this result as a consequence of our study on the intersections of incommensurable self-similar sets on R. Our methods also allow us to give upper bounds for dimensions of arbitrary slices of planar self-similar sets satisfying SSC and certain natural irreducible conditions.