Dissertations, Theses, and Capstone Projects

Date of Degree

6-2017

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Kevin O'Bryant

Committee Members

Melvyn Nathanson

Leon Karp

Ruchard Bumby

Subject Categories

Mathematics | Number Theory

Keywords

inhomogeneous Diophantine approximation, atypical numbers, fractional parts of roots

Abstract

For any positive real number $\theta > 1$, and any natural number $n$, it is obvious that sequence $\theta^{1/n}$ goes to 1. Nathanson and O'Bryant studied the details of this convergence and discovered some truly amazing properties. One critical discovery is that for almost all $n$, $\displaystyle\floor{\frac{1}{\fp{\theta^{1/n}}}}$ is equal to $\displaystyle\floor{\frac{n}{\log\theta}-\frac{1}{2}}$, the exceptions, when $n > \log_2 \theta$, being termed atypical $n$ (the set of which for fixed $\theta$ being named $\mcA_\theta$), and that for $\log\theta$ rational, the number of atypical $n$ is finite. Nathanson left a number of questions open, and, subsequently, O'Bryant developed a theory to answer most of these questions. He also posed five new unanswered questions of his own of which we completely answer three, and partially answer two. He constructed infinite sets of bounded $\theta$'s with rational logarithms, some with no atypical $n$, and some with infinitely many atypical $n$. However, he left as an open problem whether there was some upper bound, $\theta_0$ such that \newline $\{ \theta: \theta > \theta_0, \log\theta \text{ is irrational, and} \mcA_\theta \text{ is finite}}$ is not uncountable, which is his third question. This thesis shows that the restriction of boundedness cannot be removed. During the course of the development needed to answer that question, this thesis proceeds to answer other questions including demonstrating the existence of $\theta$ with logarithms that are algebraic irrational which have no atypical numbers and those which have infinite atypical sets. This thesis also shows the atypical set for $e^e$ is infinite and finds its atypical set explicitly.

Comments

A revised version was uploaded on June 18, 2018 with the approval of the Graduate Center.

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