Bounds on arithmetic projections, and applications to the Kakeya conjecture

Let $A$, $B$, be finite subsets of a torsion-free 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 $\# \{ a-b: (a,b) \in G \}$. In \cite{borg:high-dim} Bourgain obtained the bound of $N^{2-\frac{1}{13}}$, and applied this to the Kakeya conjecture. We improve Bourgain’s estimate to $N^{2-\frac{1}{6}}$, and obtain the further improvement of $N^{2 - \frac{1}{4}}$ under the additional assumption $\# \{ a+2b: (a,b) \in G\} \leq N$. As an application we conclude that Besicovitch sets in $\R^n$ have Minkowski dimension at least $\frac{4n}{7} + \frac{3}{7}$. This is new for $n > 8$.

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Published 1 January 1999