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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 Urbana-Champaign
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:2014-09-16
URI:http://hdl.handle.net/2142/50698
Rights Information:Copyright 2014 Katherine Alexander Anders
Date Available in IDEALS:2014-09-16
Date Deposited:2014-08


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