On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane

Abstract

Let $0\le s\le 1$ and $0\le t\le 2$. An $(s,t)$-Furstenberg set is a set $K\subset {\mathbb{R}}^2$ with the following property: there exists a line set $\mathcal{L}$ of Hausdorff dimension ${\dim }_{\mathrm{H}}\mathcal{L}\ge t$ such that ${\dim }_{\mathrm{H}}(K\cap \ell )\ge s$ for all $\ell \in \mathcal{L}$. We prove that for $s\in (0,1)$, and $t\in (s,2]$, the Hausdorff dimension of $(s,t)$-Furstenberg sets in ${\mathbb{R}}^2$ is no smaller than $2s+\epsilon$, where $\epsilon >0$ depends only on $s$ and $t$. For $s>1/2$ and $t=1$, this is an $\epsilon$-improvement over a result of Wolff from 1999. The same method also yields an $\epsilon$-improvement to Kaufman's projection theorem from 1968. We show that if $s\in (0,1)$, $t\in (s,2]$ and $K\subset {\mathbb{R}}^2$ is an analytic set with ${\dim }_{\mathrm{H}}K=t$, then $${\dim }_{\mathrm{H}}\{e\in S^1:{\dim }_{\mathrm{H}}{\pi }_e(K)\le s\}\le s-\epsilon ,$$ where $\epsilon >0$ only depends on $s$ and $t$. Here ${\pi }_e$ is the orthogonal projection to $\mathrm{span}(e)$.

arXiv comment64 pages, 2 figures. v3: corrected and improved Proposition 5.2. To appear in Duke Math J