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

### Tuomas Orponen

University of Helsinki, Finland

## Abstract

Let $0≤s≤1$. A set $K⊂R_{2}$ is a Furstenberg $s$-set if for every unit vector $e∈S_{1}$, 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 $dim_{H}K≥max{s+1/2,2s}$ and conjectured that $dim_{H}K≥(1+3s)/2$. The only known improvement to Wolff's bound is due to Bourgain, who proved in 2003 that $dim_{H}K≥1+ϵ$ 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: $dim_{p}K≥2s+ϵ$ for all Furstenberg $s$-sets $K⊂R_{2}$, 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 $δ>0$. 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 $K⊂R_{2}$ is a linearly measurable set with positive length, and $1/2<s<1$, then $dim_{H}{e∈S_{1}:dim_{p}π_{e}(K)≤s}≤s−ϵ$ for some $ϵ>0$ depending only on $s$. Here $π_{e}$ is the orthogonal projection onto the line spanned by $e$.

## Cite this article

Tuomas Orponen, An improved bound on the packing dimension of Furstenberg sets in the plane. J. Eur. Math. Soc. 22 (2020), no. 3, pp. 797–831

DOI 10.4171/JEMS/933