Analysis Seminar: An improved Minkowski dimension estimate for Kakeya sets in higher dimensions using planebrushes

Abstract:

A Kakeya set is defined as a compact subset of $\mathbb{R}^n$ which contains a line segment of length 1 in every direction. The Kakeya conjecture says that every Kakeya set has Minkowski and Hausdorff dimensions equal to $n$. Interest in this conjecture began around 1971, when Fefferman used Kakeya sets to construct a counterexample to the ball multiplier conjecture in Fourier analysis. Fefferman's work shows that if the Kakeya conjecture is false, other large conjectures in Fourier analysis, the Fourier restriction conjecture and the Bochner-Riesz conjecture, would be false as well.
In this talk, I will discuss an improved Minkowski dimension estimate for Kakeya sets in dimensions $n\geq 5$. The improved estimate comes from using a geometric argument called a ``$k$-planebrush'', which is a higher dimensional analogue of Wolff's ``hairbrush'' argument from 1995. The $k$-planebrush argument is used in conjunction with a previously known "k-linear" result on Kakeya sets proved by Hickman-Rogers-Zhang (and concurrently by Zahl) in 2019 along with an x-ray transform estimate which is a corollary of Hickman-Rogers-Zhang (and Zahl). The x-ray transform estimate is used to deduce that the Kakeya set has a structural property called ``stickiness,'' which was first introduced in a paper by Katz-Laba-Tao in 1999. Sticky Kakeya sets exhibit a self-similar structure which is exploited by the $k$-planebrush argument.

Host: Alan Chang