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 suchth at # A, #B,#{a + b :( a, b) ∈ G }≤ N. We consider the question of estimating the quantity #{a − b :( a, b) ∈ G}. In (2) Bourgain obtained the bound of N 2− 1 13 , and applied this to the Kakeya conjecture. We improve Bourgain's estimate to N 2− 1 6 , and obtain the further improvement of

Locations

• Mathematical Research Letters - View - PDF
• arXiv (Cornell University) - PDF
• Mathematical Research Letters - View - PDF
• arXiv (Cornell University) - PDF