From incidence geometry of lines towards incidence geometry of tubes

Larry Guth
Massachusetts Institute of Technology

In the late 90's, Tom Wolff began to adapt arguments from incidence geometry to study Kakeya-type problems in harmonic analysis. Kakeya-type problems involve large finite sets of cylinders in Euclidean space, and one wants to understand if they can have a lot of high-multiplicity intersections. The problems look very similar to incidence geometry problems with thin tubes in place of straight lines. In spite of this analogy, it's often difficult to adapt incidence geometry tools to problems about tubes.

In this talk, we will discuss adapting polynomial arguments in incidence geometry to the setting of tubes. We will recall an incidence geometry theorem proven using polynomial arguments. We will discuss the analogous problem with tubes in place of lines. We'll recall the proof using polynomial arguments, and discuss step by step the issues in adapting these arguments from lines to tubes.

Here are some of the polynomial ingredients we'll discuss: 1. finding a polynomial that vanishes at given points by linear algebra. 2. If a polynomial vanishes at many points on a line, then it vanishes on the whole line. 3. If P vanishes on three lines through a point x, and if Z(P) is smooth at x, then Z(P) is flat at x.

Here are the specific questions about lines and tubes that we will discuss. Suppose that L is a set of N^2 lines in R^3, and that X is a set of N^{3-\sigma} points, for some \sigma>0. Suppose that each line of L contains N points of X, and that each point of X lies in roughly N^\sigma lines of L. Then, if N is large enough, there is a 2-plane that contains at least N^{1+\sigma} lines of L. This theorem is a tiny variation of a result of Nets Katz and myself.

Here is an analogous problem for tubes, that is closely related to the Kakeya problem. Suppose that T is a set of N^2 cylinders in R^3 of radius 1 and length N all contained in a ball of radius N. Suppose that X is a set of N^{3 - \sigma} disjoint unit cubes in the ball, for some \sigma > 0. Suppose that each tube of T intersects N cubes of X, and that each cube of X intersects roughly N^\sigma tubes of T. Does it follow that the tubes of T cluster in a planar slab? More precisely, does it follow that for some W > 1, there is a planar slab of dimensions W x N x N that contains at least around W N^{1 + \sigma} tubes of T? This open problem was suggested by Wolff -- it's a cousing of the Kakeya problem.

Under these hypotheses and an additional technical hypothesis, we prove that there is a 1 x N^\sigma x N^\sigma slab that contains segments of length N^\sigma from at least c N^{2 \sigma} different tubes of T. This estimate can be thought of as a baby version of the open problem above.

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