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Title:  Properties of digital representations 
Author(s):  Anders, Katherine 
Director of Research:  Reznick, Bruce A. 
Doctoral Committee Chair(s):  Duursma, Iwan M. 
Doctoral Committee Member(s):  Reznick, Bruce A.; Hildebrand, A.J.; Yong, Alexander 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  number theory
combinatorics digital representations generalized binary representations 
Abstract:  Let $\mathcal{A}$ be a finite subset of $\mathbb{N}$ including $0$ and $f_\mathcal{A}(n)$ be the number of ways to write $n=\sum_{i=0}^{\infty}\epsilon_i2^i$, where $\epsilon_i\in\mathcal{A}$. The sequence $\left(f_\mathcal{A}(n)\right) \bmod 2$ is always periodic, and $f_\mathcal{A}(n)$ is typically more often even than odd. We give four families of sets $\left(\mathcal{A}_m\right)$ with $\left\mathcal{A}_m\right=4$ such that the proportion of odd $f_{\mathcal{A}_m}(n)$'s goes to $1$ as $m\to\infty$. We also consider asymptotics of the summatory function $s_\mathcal{A}(r,m)=\displaystyle\sum_{n=m2^r}^{m2^{r+1}1}f_{\mathcal{A}}(n)$ and show that $s_{\mathcal{A}}(r,m)\approx c(\mathcal{A},m)\left\mathcal{A}\right^r$ for some $c(\mathcal{A},m)\in\mathbb{Q}$. 
Issue Date:  20140916 
URI:  http://hdl.handle.net/2142/50698 
Rights Information:  Copyright 2014 Katherine Alexander Anders 
Date Available in IDEALS:  20140916 
Date Deposited:  201408 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois