Abstract

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.