Publication Date

Fall 12-13-2025

Abstract

The Hausdorff dimension of Kakeya sets is an interesting problem where the two-dimensional case can be proven directly, but even obtaining bounds on the higher dimension analogues can require highly technical machinery. The difficulty of the general case has inspired analysts to look at Kakeya sets from non-Euclidean viewpoints.

In this paper, we explore a construction of the Kakeya set in the first Heisenberg group H^1. By utilizing the sub-Riemannian manifold of H^1 we can apply tools in geometric measure theory which at this time cannot be applied in R^3. We restate and provide a detailed proof for a sharp bound obtained by Jiayin Liu for these so-called Kakeya-Heisenberg sets. We also discuss recent results in other non-Euclidean spaces and in R^3.

Degree Name

Mathematics

Level of Degree

Masters

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Dimiter Vassilev

Second Committee Member

Daniel Gomez

Third Committee Member

Hongnian Huang

Fourth Committee Member

Annunziata Loiudice

Language

English

Keywords

kakeya heisenberg geometric measure theory hausdorff

Document Type

Thesis

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Mathematics Commons

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