  Let $a < b$ be multiplicatively independent integers, both at least $2$. Let
$A,B$ be closed subsets of $[0,1]$ that are forward invariant under
multiplication by $a$, $b$ respectively, and let $C := A\times B$. An old
conjecture of Furstenberg asserted that any planar line $L$ not parallel to
either axis must intersect $C$ in Hausdorff dimension at most $\max\{\dim C,1\}
- 1$. Two recent works by Shmerkin and Wu have given two different proofs of
this conjecture. This note provides a third proof. Like Wu's, it stays close to
the ergodic theoretic machinery that Furstenberg introduced to study such
questions, but it uses less substantial background from ergodic theory. The
same method is also used to re-prove a recent result of Yu about certain
sequences of sums.
