Many problems in geometric measure theory are centered around finding conditions and structures on a set to guarantee that its distance set must be large. Two notions of structure that are of importance in this work are Hausdorff dimension and thickness. Recent progress has been made on generalizing the notion of thickness so part of this work also generalizes previous results using this new upgraded version of thickness. We also show why a famous conjecture about distance sets does not hold on the real line and thus, why this conjecture needs to happen in higher dimensions. Furthermore, we give explicit distance set and thickness calculations for a special class of self-similar sets. / Master of Science / Part of the study of geometric measure theory is centered around creating interesting structures to place on a set and determining what sort of threshold on that structure allows you to guarantee that some interesting geometric property exists for that set. An example of this is determining when you can guarantee that a set contains many unique distances between elements in that set. This work presents various types of structures that help to investigate the problem of when you can guarantee that a set has the previously mentioned geometric property.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/110350 |
| Date | 26 May 2022 |
| Creators | Boone, Zackary Ryan |
| Contributors | Mathematics, Palsson, Eyvindur Ari, Yang, Yun, Sun, Wenbo |
| Publisher | Virginia Tech |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | ETD, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
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