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A Kakeya Estimate for Sticky Sets Using a Planebrush


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.


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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Kakeya conjecture, Besicovitch set, sticky Kakeya sets
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:Apostol Award for Excellence in Teaching in Mathematics, 2021, 2023.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Conlon, David
Thesis Committee:
  • Graber, Thomas B. (chair)
  • Conlon, David
  • Isett, Philip
  • Katz, Nets H.
Defense Date:7 June 2024
Record Number:CaltechTHESIS:06102024-225449252
Persistent URL:
Kulkarni, Neeraja Raghavendra0000-0001-7747-9177
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:16515
Deposited By: Neeraja Kulkarni
Deposited On:11 Jun 2024 22:02
Last Modified:18 Jun 2024 19:09

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