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

Abstract

We prove the following conjecture of Furstenberg (1969): if $A,B\subset [0,1]$ are closed and invariant under $\times p\mathrm{mod}1$ and $\times q\mathrm{mod}1$, respectively, and if $\log p/\log q\notin \mathbb{Q}$, then for all real numbers $u$ and $v$, $${\dim }_{\mathrm{H}}(uA+v)\cap B\le \max \{0,{\dim }_{\mathrm{H}}A+{\dim }_{\mathrm{H}}B-1\}.$$ 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.

37 pages, v2: Typos fixed, Other small changes; v3: incorporates referee's suggestions, many small fixes, main results unchanged, To appear in Annals of Math

References in corpus (2)

• Equidistribution from Fractals
• Slicing the Sierpiński gasket