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

Abstract

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 ${\mathbb{R}}^n$ have Hausdorff dimension at least 6n/11+5/11 and Minkowski dimension at least $4n/7+3/7$. This is new for $n>8$.

6 pages, submitted to Math Research Letters; improved bounds in revised version; typoes corrected in second revised version