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

TL;DR

This paper improves lower bounds on the Hausdorff dimension of Furstenberg sets and the size of exceptional sets for orthogonal projections in the plane, advancing understanding of geometric measure theory.

Contribution

It provides new epsilon-improvements on Hausdorff dimension bounds for Furstenberg sets and projection theorems, extending classical results.

Findings

Hausdorff dimension of (s,t)-Furstenberg sets is at least 2s + epsilon

Improves Wolff's 1999 result for s > 1/2 and t=1

Reduces the size of the exceptional set for projections of sets with given Hausdorff dimension

Abstract

Let $0 \leq s \leq 1$ and $0 \leq t \leq 2$. 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 $\dim_{\mathrm{H}} \mathcal{L} \geq t$ such that $\dim_{\mathrm{H}} (K \cap \ell) \geq s$ 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 $$\dim_{\mathrm{H}} \{e \in S^{1} : \dim_{\mathrm{H}} \pi_{e}(K) \leq s\} \leq s - \epsilon,$$ where $\epsilon > 0$ only depends on $s$ and $t$. Here $\pi_{e}$ is the orthogonal projection to $\mathrm{span}(e)$.