Properties of digital representations

number theory;combinatorics;digital representations;generalized binary representations

Anders, Katherine

关键词: number theory;    combinatorics;    digital representations;    generalized binary representations;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/50698/Katherine_Anders.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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 $c(\mathcal{A},m)\in\mathbb{Q}$.