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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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