Automata and Ratio Sets

Abstract

This thesis explores the composition of ratio sets, the subsets of the rationals derived from the quotients of two sets of natural numbers, and examines a variety of specific examples where the comprising sets of natural numbers have specific properties. I present a general algorithm that decides the inclusion of a rational number in a specific ratio set if the comprising sets of natural numbers are a regular language when represented in a given base. I also present an algorithm for deciding the inclusion of a rational number in the ratio set of a few select sets of natural numbers that are not a regular language when represented in any base, namely, the set of natural numbers with representations in a specific base that are palindromes or antipalindromes. Using those algorithms, I examine some of the rational numbers in specific ratio sets and then prove several results regarding the composition of those ratio sets. As well, I present algorithms for computing approximations to real numbers using elements of some specific ratio sets.