Diophantine Avoidance, Number Fields, and Quadratic Forms

<p>Diophantine avoidance has been studied by several authors in recent years. This term refers to effective results on existence of points of bounded size (where size is measured by norm or height, depending on the context) in a given algebraic set avoiding some specified subsets. The application of avoidance conditions allows to understand how ``well distributed" are points of bounded size in a given set. If it is possible to find them outside of some prescribed collection of subsets of the set in question, then it suggests that they are evenly distributed, in some appropriate sense. Our first result investigates small-norm points in lattices with avoidance conditions outside of a hypersurface of arbitrary degree. The main application of this investigation is to small-height generators of number fields satisfying certain natural avoidance conditions. Further, we study small-size integer zeros of integral quadratic forms with avoidance conditions. We apply our results to the problem of effective distribution of angles between vectors in Z n .</p>

Name thesis:degree_name

Mathematics, PhD

Level thesis:degree_level

Open Access Dissertation

Discipline thesis:degree_discipline

Institute of Mathematical Sciences

Year dc:date.available

2025

Author dc:creator

• Jeong, Sehun

Contributors dc:contributor

• Allon Percus
• Helen Wong

• Geometry of numbers
• Height functions
• Lattice theory
• Number fields
• Number theory
• Quadratic forms
• Applied Mathematics
• Mathematics

Repository record dc:identifier

https://scholarship.claremont.edu/cgu_etd/1020

OAI identifier oai:identifier

oai:scholarship.claremont.edu:cgu_etd-2042