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

Let 0 ≤ s ≤ 1 . A set K ⊂ R² is a Furstenberg s -set if for every unit vector e ∈ S¹ , some line Lₑ parallel to e satisfies _H [K ∩ Lₑ] ≥ 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 _H K ≥ max\{s + 1/2,2s\} and conjectured that _H K ≥ (1 + 3s)/2 . The only known improvement to Wolff's bound is due to Bourgain, who proved in 2003 that _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: _p K ≥ 2s + ε for all Furstenberg s -sets K ⊂ R² , 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 ⊂ R² is a linearly measurable set with positive length, and 1/2 < s < 1 , then _H \{e ∈ S¹ : _p πₑ(K) ≤ s\} ≤ s - ε for some ε > 0 depending only on s . Here πₑ is the orthogonal projection onto the line spanned by e .