#### Date of Degree

2-2014

#### Document Type

Dissertation

#### Degree Name

Ph.D.

#### Program

Mathematics

#### Advisor(s)

Melvyn Nathanson

#### Subject Categories

Mathematics

#### Keywords

diophantine equations, lambda sequence, metric diameter, special g-adic partitions

#### Abstract

{Let $X$ be a group with identity $e$, we define $A$ as an infinite set of generators for $X$, and let $(X,d)$ be the metric space with word length $d_{A}$ induced by $A$. Nathanson showed that if $P$ is a nonempty finite set of prime numbers and $A$ is the set of positive integers whose prime factors all belong to $P$, then the metric space $({\bf{Z}},d_{A})$ has infinite diameter. Nathanson also studied the $\lambda_{A}(h)$ sequences, where $\lambda_{A}(h)$ is defined as the smallest positive integer $y$ with $d_{A}(e,y)=h$, and he posed the problem to compute $\lambda_{A}(h)$ and estimate its growth rate. We will give explicit forms for $\lambda_{p}(h)$ for any fixed odd integer $p>1$. We will also solve the open problems of computing the term $\lambda_{2,3}(4)$, provide an explicit lower bound for $\lambda_{2,3}(h)$ and classifying $\lambda_{2,p}(h)$ for $p>1$ any odd integer and $h\in\{1,2,3\}$. }

#### Recommended Citation

Singh, Satyanand, "Special Representations, Nathanson's Lambda Sequences and Explicit Bounds" (2014). *CUNY Academic Works.*

https://academicworks.cuny.edu/gc_etds/112