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

Let 0 <= s <= 1. A set K subset of R-2 is a Furstenberg s-set if for every unit vector e is an element of S-1, some line L-e parallel to e satisfies dim(H) [K boolean AND L-e] >= s. The Furstenberg set problem, introduced by T. Wolff in 1999, asks about 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 + 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 >= 2s + epsilon for all Furstenberg s-sets K subset of 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 of R-2 is a linearly measurable set with positive length, and 1/2 < s < 1, then dim(H) {e is an element of S-1 : dim(p) pi(e)(K) <= s} <= s - epsilon for some epsilon > 0 depending only on s. Here pi(e) is the orthogonal projection onto the line spanned by e.