A Kakeya Estimate for Sticky Sets Using a Planebrush

Author: Kulkarni, Neeraja Raghavendra

Year: 2024

Degree: Dissertation (Ph.D.)

Advisor: Conlon, David

Committee Members: Graber, Thomas B.; Conlon, David; Isett, Philip; Katz, Nets H.

Option: Mathematics

DOI: 10.7907/japt-b214

Abstract

A Besicovitch set is defined as a compact subset of ℝⁿ which contains a line segment of length 1 in every direction. The Kakeya conjecture says that every Besicovitch set has Minkowski and Hausdorff dimensions equal to n. This thesis gives an improved Hausdorff dimension estimate, d ⩾ 0.60376707287 n + O(1), for Besicovitch sets displaying a special structural property called "stickiness." The improved estimate comes from using an incidence geometry argument called a "k-planebrush," which is a higher dimensional analogue of Wolff's "hairbrush" argument from 1995.

In addition, an x-ray transform estimate is obtained as a corollary of Zahl's k-linear estimate in 2019. The x-ray estimate, together with the estimate for sticky sets, implies that all Besicovitch sets in ℝⁿ must have Minkowski dimension greater than (2 - √2 + ε)n. Though this Minkowski dimension estimate is not as good as one previously known from Katz-Tao(2000), it provides a new proof of the same result.

Files