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

TL;DR

This paper improves bounds on the size of difference-sets in finite abelian groups, leading to new lower bounds on the Hausdorff and Minkowski dimensions of Besicovitch sets in high-dimensional Euclidean spaces, with applications to the Kakeya conjecture.

Contribution

It advances the bounds on difference-set sizes in abelian groups and applies these results to improve dimension estimates for Besicovitch sets, impacting the Kakeya conjecture.

Findings

Improved the bound on difference-sets to N^{2-1/6}.

Further improved the bound to N^{2-1/4} under additional conditions.

Established new lower bounds for Hausdorff and Minkowski dimensions of Besicovitch sets in dimensions greater than 8.

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