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 $n > 8$.
Categories: math.CO math.CA