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 $G \subset A \times B$besuchthat$# A, # B, # {a+b: (a,b) \in G } \leq N$. 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 $N^{2-1\mathrm{/}13}$, and applied this to the Kakeya conjecture. We improve Bourgain’s estimate further to $N^{2-1\mathrm{/}6}$, and obtain the further improvement of $N^{2-1\mathrm{/}4}$ if we also know that # \{a+2b: (a,b) \in G\} \leq N. We conclude that Besicovitch sets in ${\mathbb{R}}^n$ have Hausdorff dimension at least 6n/11+5/11 and Minkowski dimension at least $4n\mathrm{/}7+3\mathrm{/}7$. This is new for $n>8$.