The Mattila-Sjölin Problem for Triangles

This dissertation contains work from the author's papers [35] and [36] with coauthor Eyvindur Palsson. The classic Mattila-Sjolin theorem shows that if a compact subset of mathbb{R}^d has Hausdorff dimension at least $frac{(d+1)}{2}$ then its set of distances has nonempty interior. In this dissertation, we present a similar result, namely that if a compact subset $E$ of mathbb{R}^d, with $d geq 3$, has a large enough Hausdorff dimension then the set of congruence classes of triangles formed by triples of points of $E$ has nonempty interior. These types of results on point configurations with nonempty interior can be categorized as extensions and refinements of the statement in the well known Falconer distance problem which establishes a positive Lebesgue measure for the distance set instead of it having nonempty interior

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

2023

Author dc:creator

• Romero Acosta, Juan Francisco

Chair dc:contributor.committeechair

• Palsson, Eyvindur Ari

Committee members dc:contributor.committeemember

• Yang, Yun
• Sun, Wenbo
• Elgart, Alexander

• harmonic analysis
• geometric measure theory
• Hausdorff dimension

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

en

Dc Identifier Other

vt_gsexam:36838

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/114979