A new bound on partial sum-sets and difference-sets, and applications to the Kakeya conjecture

Let $A, B$, be finite subsets of an abelian group, and let $G \subset A \times B$ be such that # A, # B, # \{a+b: (a,b) \in G \} \leq N. We consider the question of estimating the quantity # \{a-b: (a,b) \in G \}. Recently Bourgain improved the trivial upper bound of $N^2$ to $N^{2-1/13}$, and applied this to the Kakeya conjecture. We improve Bourgain's estimate further to $N^{2-1/6}$, and obtain the further improvement of $N^{2-1/4}$ if we also know that # \{a+2b: (a,b) \in G\} \leq N. We conclude that Besicovitch sets in $\R^n$ have Hausdorff dimension at least 6n/11+5/11 and Minkowski dimension at least $4n/7 + 3/7$. This is new for $n > 8$.