A Kakeya Estimate for Sticky Sets Using a Planebrush

Citation

Kulkarni, Neeraja Raghavendra (2024) A Kakeya Estimate for Sticky Sets Using a Planebrush. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/japt-b214. https://resolver.caltech.edu/CaltechTHESIS:06102024-225449252

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.

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