Bounds on arithmetic projections, and applications to the Kakeya conjecture

Abstract

Let A, B, be finite subsets of a torsion-free abelian group, and let G ⊂ A × B be such that #A, #B, #{a + b : (a, b) ∈ G} ≤ N. We consider the question of estimating the quantity #{a - 6 : (a, b) ∈ G}. In [2] Bourgain obtained the bound of N2-1/13, and applied this to the Kakeya conjecture. We improve Bourgain's estimate to N2-1/6, and obtain the further improvement of N2-1/4 under the additional assumption #{a + 2b : (a, b) ∈ G} ≤ N. As an application we conclude that Besicovitch sets in ℝ n have Minkowski dimension at least 4n/7 + 3/7. This is new for n > 8.