Date of Award

2025

Degree Type

Open Access Dissertation

Degree Name

Mathematics, PhD

Program

Institute of Mathematical Sciences

Advisor/Supervisor/Committee Chair

Lenny Fukshansky

Dissertation or Thesis Committee Member

Allon Percus

Dissertation or Thesis Committee Member

Helen Wong

Terms of Use & License Information

Terms of Use for work posted in Scholarship@Claremont.

Rights Information

© 2025 Sehun Jeong

Keywords

Geometry of numbers, Height functions, Lattice theory, Number fields, Number theory, Quadratic forms

Subject Categories

Applied Mathematics | Mathematics

Abstract

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 .

ISBN

9798291577592

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