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

We prove the following conjecture of Furstenberg (1969): if $A,B\subset [0,1]$areclosedandinvariantunder$\times p \mod 1$and$\times q \mod 1$, respectively, and if $\log p\mathrm{/}\log q\mathrm{\notin }\mathbb{Q}$, then for all real numbers $u$ and $v$, $${\dim }_{\mathrm{H}}(uA+v)\cap B\le \max \{0,{\dim }_{\mathrm{H}}A+{\dim }_{\mathrm{H}}B-1\}\mathrm{.}$$ We obtain this result as a consequence of our study on the intersections of incommensurable self-similar sets on $\mathbb{R}$. 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.