  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 $n > 8$.
