The Endpoint Case of the Bennett-carbery-tao Multilinear Kakeya Conjecture

We prove the endpoint case of the multilinear Kakeya conjecture of Bennett, Carbery, and Tao. The proof uses the polynomial method introduced by Dvir. In [1], Bennett, Carbery, and Tao formulated a multilinear Kakeya conjecture, and they proved the conjecture except for the endpoint case. In this paper, we slightly sharpen their result by proving the endpoint case of the conjecture. Our method of proof is very different from the proof of Bennett, Carbery, and Tao. The original proof was based on monotonicity estimates for heat flows. In 2007, Dvir [2] made a breakthrough on the Kakeya problem, proving the Kakeya conjecture over finite fields. His proof used polynomials in a crucial way. It was not clear whether Dvir’s approach could be adapted to prove estimates in Euclidean space. Our proof of the multilinear Kakeya conjecture is based on Dvir’s polynomial method. In my opinion, the method of proof is as interesting as the result. The multilinear Kakeya conjecture concerns the overlap properties of cylindrical tubes in R. Roughly, the (multilinear) Kakeya conjecture says that cylinders pointing in different directions cannot overlap too much. Before coming to the Bennett-Carbery-Tao multilinear estimate, I want to state a weaker result, because it’s easier to understand and easier to prove. To be clear about the notation, a cylinder of radius R around a line L ⊂ R is the set of all points x ∈ R within a distance R of the line L. We call the line L the core of the cylinder. Theorem 1. Suppose we have a finite collection of cylinders Tj,a ⊂ R, where 1 ≤ j ≤ n, and 1 ≤ a ≤ A for some integer A. Each cylinder has radius 1. Moreover, each cylinder Tj,a runs nearly parallel to the xj-axis. More precisely, we assume that the angle between the core of Tj,a and the xj-axis is at most (100n) . We let I be the set of points that belong to at least one cylinder in each direction. In symbols, I := ∩j=1 [ ∪a=1Tj,a ] . Then V ol(I) ≤ C(n)A n n−1 . As Bennett, Carbery, and Tao point out in [1], this estimate can be viewed as a generalization of the Loomis-Whitney inequality. Theorem. (special case of Loomis and Whitney, 1949, [11]) Let U be an open set in R. Let πj denote the projection from R n onto the hyperplane perpendicular to the xj-axis. Suppose that for each j, πj(U) has (n-1)-dimensional volume at most B. Then V ol(U) ≤ B n n−1 . 1