We prove the following conjecture of Furstenberg (1969): if ๐ด,๐ตโ[0,1] are closed and invariant under ร๐ mod1 and ร๐ mod1, respectively, and if log๐/log๐โโ, then for all real numbers ๐ข and ๐ฃ,
dim_H (๐ข๐ด + ๐ฃ) โฉ ๐ต โค max {0, dim_H ๐ด + dim_H ๐ต โ 1}.
We obtain this result as a consequence of our study on the intersections of incommensurable self-similar sets on โ. 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.