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 $2s+\mathit{ϵ}$, where $\mathit{ϵ}>0$ depends only on s and t. For $s>1∕2$ and $t=1$, this is an ϵ-improvement over a result of Wolff from 1999.

The same method also yields an ϵ-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

$${dim}_{\mathrm{H}}\{e\in S^1:{dim}_{\mathrm{H}}{\mathrm{\pi }}_e(K)\le s\}\le s-\mathit{ϵ},$$

where $\mathit{ϵ}>0$ depends only on s and t. Here ${\mathrm{\pi }}_e$ is the orthogonal projection to the line in direction e.