A new dynamical proof of the Shmerkin–Wu theorem

Abstract

Let \begin{document}$ a < b $\end{document} be multiplicatively independent integers, both at least \begin{document}$ 2 $\end{document}. Let \begin{document}$ A,B $\end{document} be closed subsets of \begin{document}$ [0,1] $\end{document} that are forward invariant under multiplication by \begin{document}$ a $\end{document}, \begin{document}$ b $\end{document} respectively, and let \begin{document}$ C : = A\times B $\end{document}. An old conjecture of Furstenberg asserted that any planar line \begin{document}$ L $\end{document} not parallel to either axis must intersect \begin{document}$ C $\end{document} in Hausdorff dimension at most \begin{document}$ \max\{\dim C,1\} - 1 $\end{document}. 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.