Let . A set is a Furstenberg -set if for every unit vector , some line parallel to satisfies
The Furstenberg set problem, introduced by T. Wolff in 1999, asks for the best lower bound for the dimension of Furstenberg -sets. Wolff proved that and conjectured that . The only known improvement to Wolff's bound is due to Bourgain, who proved in 2003 that for Furstenberg -sets , where is an absolute constant. In the present paper, I prove a similar -improvement for all , but only for packing dimension: for all Furstenberg -sets , where only depends on .
The proof rests on a new incidence theorem for finite collections of planar points and tubes of width . 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 is a linearly measurable set with positive length, and , then for some depending only on . Here is the orthogonal projection onto the line spanned by .
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. 797–831DOI 10.4171/JEMS/933