## All Dissertations, Theses, and Capstone Projects

6-2017

Dissertation

Ph.D.

Mathematics

Kevin O'Bryant

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.