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 subs et of A x B be such that #A, #B, #{a + b : (a, b) is an element of G} less than or equal to N. We consider the question of estimating the quantity #{a - b : (a, b) is an element of G}. In [2] Bourgain obtained the bound of N2 -1/13, and applied this to the Kakeya conjecture. We improve Bourgain's est imate to N2-1/6, and obtain the further improvement of N2-1/4 under the add itional assumption #{a + 2b : (a, b) is an element of G} less than or equal to N. As an application we conclude that Besicovitch sets in an have Minko wski dimension at least 4n/7 + 3/7. This is new for n > 8.