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

  • Tuomas Orponen

    University of Helsinki, Finland
An improved bound on the packing dimension of Furstenberg sets in the plane cover
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Abstract

Let 0s10 \leq s \leq 1. A set KR2K \subset \mathbb{R}^{2} is a Furstenberg ss-set if for every unit vector eS1e \in S^{1}, some line LeL_{e} parallel to ee satisfies

dimH[KLe]s.\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 ss-sets. Wolff proved that dimHKmax{s+1/2,2s}\dim_H K \geq \max\{s + 1/2,2s\} and conjectured that dimHK(1+3s)/2\dim_H K \geq (1 + 3s)/2. The only known improvement to Wolff's bound is due to Bourgain, who proved in 2003 that dimHK1+ϵ\dim_H K \geq 1 + \epsilon for Furstenberg 1/21/2-sets KK, where ϵ>0\epsilon > 0 is an absolute constant. In the present paper, I prove a similar ϵ\epsilon-improvement for all 1/2<s<11/2 < s < 1, but only for packing dimension: dimpK2s+ϵ\dim_p K \geq 2s + \epsilon for all Furstenberg ss-sets KR2K \subset \mathbb{R}^{2}, where ϵ>0\epsilon > 0 only depends on ss.

The proof rests on a new incidence theorem for finite collections of planar points and tubes of width δ>0\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 KR2K \subset \mathbb{R}^{2} is a linearly measurable set with positive length, and 1/2<s<11/2 < s < 1, then

dimH{eS1:dimpπe(K)s}sϵ\dim_H \{e \in S^{1} : \dim_p \pi_{e}(K) \leq s\} \leq s - \epsilon

for some ϵ>0\epsilon > 0 depending only on ss. Here πe\pi_{e} is the orthogonal projection onto the line spanned by ee.

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