[
{
"id": "thesis:16515",
"collection": "thesis",
"collection_id": "16515",
"cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06102024-225449252",
"primary_object_url": {
"basename": "Kulkarni_Neeraja_2024.pdf",
"content": "final",
"filesize": 521723,
"license": "other",
"mime_type": "application/pdf",
"url": "/16515/1/Kulkarni_Neeraja_2024.pdf",
"version": "v4.0.0"
},
"type": "thesis",
"title": "A Kakeya Estimate for Sticky Sets Using a Planebrush",
"author": [
{
"family_name": "Kulkarni",
"given_name": "Neeraja Raghavendra",
"orcid": "0000-0001-7747-9177",
"clpid": "Kulkarni-Neeraja-Raghavendra"
}
],
"thesis_advisor": [
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
}
],
"thesis_committee": [
{
"family_name": "Graber",
"given_name": "Thomas B.",
"clpid": "Graber-T-B"
},
{
"family_name": "Conlon",
"given_name": "David",
"orcid": "0000-0001-5899-1829",
"clpid": "Conlon-David"
},
{
"family_name": "Isett",
"given_name": "Philip",
"orcid": "0000-0001-9038-5546",
"clpid": "Isett-Phlip"
},
{
"family_name": "Katz",
"given_name": "Nets H.",
"orcid": "0000-0002-6239-5429",
"clpid": "Katz-N-H"
}
],
"local_group": [
{
"literal": "div_pma"
}
],
"abstract": "A Besicovitch set is defined as a compact subset of \u211d\u207f 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 \u2a7e 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.

\r\n \r\nIn 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 \u211d\u207f must have Minkowski dimension greater than (2 - \u221a2 + \u03b5)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.

",
"doi": "10.7907/japt-b214",
"publication_date": "2024",
"thesis_type": "phd",
"thesis_year": "2024"
}
]