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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2306.01770 |
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| _version_ | 1866916958427938816 |
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| author | Takamizo, Fumichika |
| author_facet | Takamizo, Fumichika |
| contents | Let $β>1$. For $x \in [0,\infty)$, we have so-called the $β$-expansion of $x$ in base $β$ as follows: $$x= \sum_{j \leq k} x_{j}β^{j} = x_{k}β^{k}+ \cdots + x_{1}β+x_{0}+x_{-1}β^{-1} + x_{-2}β^{-2} + \cdots$$ where $k \in \mathbb{Z}$, $β^{k} \leq x < β^{k+1}$, $x_{j} \in \mathbb{Z} \cap [0,β)$ for all $j \leq k$ and $\sum_{j \leq n}x_{j}β^{j}<β^{n+1}$ for all $n \leq k$. In this paper, we give a sufficient condition (for $β$) such that each element of $\mathbb{N}$ has the finite beta-expansion in base $β$. Moreover we also find a $β$ with this finiteness property which does not have positive finiteness property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_01770 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Finite beta-expansions of natural numbers Takamizo, Fumichika Number Theory Let $β>1$. For $x \in [0,\infty)$, we have so-called the $β$-expansion of $x$ in base $β$ as follows: $$x= \sum_{j \leq k} x_{j}β^{j} = x_{k}β^{k}+ \cdots + x_{1}β+x_{0}+x_{-1}β^{-1} + x_{-2}β^{-2} + \cdots$$ where $k \in \mathbb{Z}$, $β^{k} \leq x < β^{k+1}$, $x_{j} \in \mathbb{Z} \cap [0,β)$ for all $j \leq k$ and $\sum_{j \leq n}x_{j}β^{j}<β^{n+1}$ for all $n \leq k$. In this paper, we give a sufficient condition (for $β$) such that each element of $\mathbb{N}$ has the finite beta-expansion in base $β$. Moreover we also find a $β$ with this finiteness property which does not have positive finiteness property. |
| title | Finite beta-expansions of natural numbers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2306.01770 |