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

Abstract

Let $0\le s\le 1$. 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

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 ${\dim }_{H}K\ge \max \{s+1/2,2s\}$ and conjectured that ${\dim }_{H}K\ge (1+3s)/2$. The only known improvement to Wolff's bound is due to Bourgain, who proved in 2003 that ${\dim }_{H}K\ge 1+ϵ$ for Furstenberg $1/2$-sets $K$, where $ϵ>0$ is an absolute constant. In the present paper, I prove a similar $ϵ$-improvement for all $1/2<s<1$, but only for packing dimension: ${\dim }_{p}K\ge 2s+ϵ$ for all Furstenberg $s$-sets $K\subset {\mathbb{R}}^2$, where $ϵ>0$ 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 ${\dim }_H\{e\in S^1:{\dim }_p{\pi }_e(K)\le s\}\le s-ϵ$ for some $ϵ>0$ depending only on $s$. Here ${\pi }_e$ is the orthogonal projection onto the line spanned by $e$.