An improved bound on the packing dimension of Furstenberg sets in the plane

Abstract

Let $0 \leq s \leq 1$. A set $K \subset \mathbb{R}^{2}$ is a Furstenberg $s$-set, if for every unit vector $e \in S^{1}$, some line $L_{e}$ parallel to $e$ satisfies $$\dim_{\mathrm{H}} [K \cap L_{e}] \geq s.$$ The Furstenberg set problem, introduced by T. Wolff in 1999, asks for the best lower bound for the dimension of Furstenberg $s$-sets. Wolff proved that $\dim_{\mathrm{H}} K \geq \max\{s + 1/2,2s\}$ and conjectured that $\dim_{\mathrm{H}} K \geq (1 + 3s)/2$. The only known improvement to Wolff's bound is due to Bourgain, who proved in 2003 that $\dim_{\mathrm{H}} K \geq 1 + \epsilon$ for Furstenberg $1/2$-sets $K$, where $\epsilon > 0$ is an absolute constant. In the present paper, I prove a similar $\epsilon$-improvement for all $1/2 < s < 1$, but only for packing dimension: $\dim_{\mathrm{p}} K \geq 2s + \epsilon$ for all Furstenberg $s$-sets $K \subset \mathbb{R}^{2}$, where $\epsilon > 0$ only depends on $s$. The proof rests on a new incidence theorem for finite collections of planar points and tubes of width $\delta > 0$. As another corollary of this theorem, I obtain a small improvement for Kaufman's estimate from 1968 on the dimension of exceptional sets of orthogonal projections. Namely, I prove that if $K \subset \mathbb{R}^{2}$ is a linearly measurable set with positive length, and $1/2 < s < 1$, then $$\dim_{\mathrm{H}} \{e \in S^{1} : \dim_{\mathrm{p}} \pi_{e}(K) \leq s\} \leq s - \epsilon$$ for some $\epsilon > 0$ depending only on $s$. Here $\pi_{e}$ is the orthogonal projection onto the line spanned by $e$.