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_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_H K \geq \max\{s + 1/2,2s\} and conjectured that \dim_H K \geq (1 + 3s)/2 . The only known improvement to Wolff's bound is due to Bourgain, who proved in 2003 that \dim_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_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_H \{e \in S^{1} : \dim_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 .

Locations

• arXiv (Cornell University) - View - PDF
• Journal of the European Mathematical Society - View