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

TL;DR

This paper proves Furstenberg's conjecture on the Hausdorff dimension of intersections of certain invariant sets under multiplicative maps, using self-similar set intersection theory.

Contribution

It provides a proof of Furstenberg's conjecture for incommensurable invariant sets and introduces methods to bound dimensions of slices of planar self-similar sets.

Findings

Confirmed Furstenberg's conjecture for  and  with ,  invariant under ,  maps.

Derived upper bounds for dimensions of slices of self-similar sets.

Developed new techniques for analyzing intersections of incommensurable self-similar sets.

Abstract

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