Rational Points in Translations of The Cantor Set
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866929371356332032 |
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| author | Jiang, Kan Kong, Derong Li, Wenxia Wang, Zhiqiang |
| author_facet | Jiang, Kan Kong, Derong Li, Wenxia Wang, Zhiqiang |
| contents | Given two coprime integers $p\ge 2$ and $q \ge 3$, let $D_p\subset[0,1)$ consist of all rational numbers which have a finite $p$-ary expansion, and let $$ K(q, \mathcal{A})=\bigg\{ \sum_{i=1}^\infty \frac{d_i}{q^i}: d_i\in \mathcal{A}~ \forall i\in\mathbb{N} \bigg\}, $$ where $\mathcal{A} \subset \{0,1,\ldots, q-1\}$ with cardinality $1<\#\mathcal{A}< q$. In 2021 Schleischitz showed that $\#(D_p\cap K(q,\mathcal{A}))<+\infty$. In this paper we show that for any $r\in\mathbb{Q}$ and for any $α\in\mathbb{R}$, $$ \#\big((r D_p+α)\cap K(q,\mathcal{A})\big)<+\infty. $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_04624 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Rational Points in Translations of The Cantor Set Jiang, Kan Kong, Derong Li, Wenxia Wang, Zhiqiang Number Theory Dynamical Systems Primary: 11A63 Given two coprime integers $p\ge 2$ and $q \ge 3$, let $D_p\subset[0,1)$ consist of all rational numbers which have a finite $p$-ary expansion, and let $$ K(q, \mathcal{A})=\bigg\{ \sum_{i=1}^\infty \frac{d_i}{q^i}: d_i\in \mathcal{A}~ \forall i\in\mathbb{N} \bigg\}, $$ where $\mathcal{A} \subset \{0,1,\ldots, q-1\}$ with cardinality $1<\#\mathcal{A}< q$. In 2021 Schleischitz showed that $\#(D_p\cap K(q,\mathcal{A}))<+\infty$. In this paper we show that for any $r\in\mathbb{Q}$ and for any $α\in\mathbb{R}$, $$ \#\big((r D_p+α)\cap K(q,\mathcal{A})\big)<+\infty. $$ |
| title | Rational Points in Translations of The Cantor Set |
| topic | Number Theory Dynamical Systems Primary: 11A63 |
| url | https://arxiv.org/abs/2204.04624 |