Rational points in Cantor sets in the complex plane
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| Format: | Preprint |
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2025
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| _version_ | 1866912753536466944 |
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| author | Li, Wenxia Wang, Zhiqiang Zhao, Jiuzhou |
| author_facet | Li, Wenxia Wang, Zhiqiang Zhao, Jiuzhou |
| contents | Let $K$ be an imaginary quadratic field and let $\mathcal{O}_K$ be the ring of algebraic integers of $K$. For $α\in \mathcal{O}_K$ with $|α| > 1$, define \[ \mathcal{D}_α= \bigcup_{n=0}^\infty \frac{\mathcal{O}_K}{α^n}. \] For $β\in \mathcal{O}_K$ with $|β|>1$ and a finite subset $A \subset \mathcal{O}_K$, define \[ S_{β,A} = \bigg\{ \sum_{k=1}^{\infty} \frac{a_k}{β^k}: \; a_k \in A \;\forall k \in \mathbb{N} \bigg\}. \] Suppose that $α$ and $β$ are relatively prime. In this paper, we show that if $\dim_{\mathrm{H}} S_{β,A} < 1$, then the intersection $\mathcal{D}_α\cap S_{β,A}$ is a finite set. In general, the threshold for the Hausdorff dimension of $S_{β,A}$ is sharp. If we further assume that $\mathcal{O}_K$ is a unique factorization domain and that $\overlineα$ and $α$ are relatively prime, then we establish the finiteness of the intersection under the weaker condition $\dim_{\mathrm{H}} S_{β,A} < 2$. This extends the previously known results on the real line. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_07139 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rational points in Cantor sets in the complex plane Li, Wenxia Wang, Zhiqiang Zhao, Jiuzhou Number Theory Classical Analysis and ODEs 11A63, 28A80 Let $K$ be an imaginary quadratic field and let $\mathcal{O}_K$ be the ring of algebraic integers of $K$. For $α\in \mathcal{O}_K$ with $|α| > 1$, define \[ \mathcal{D}_α= \bigcup_{n=0}^\infty \frac{\mathcal{O}_K}{α^n}. \] For $β\in \mathcal{O}_K$ with $|β|>1$ and a finite subset $A \subset \mathcal{O}_K$, define \[ S_{β,A} = \bigg\{ \sum_{k=1}^{\infty} \frac{a_k}{β^k}: \; a_k \in A \;\forall k \in \mathbb{N} \bigg\}. \] Suppose that $α$ and $β$ are relatively prime. In this paper, we show that if $\dim_{\mathrm{H}} S_{β,A} < 1$, then the intersection $\mathcal{D}_α\cap S_{β,A}$ is a finite set. In general, the threshold for the Hausdorff dimension of $S_{β,A}$ is sharp. If we further assume that $\mathcal{O}_K$ is a unique factorization domain and that $\overlineα$ and $α$ are relatively prime, then we establish the finiteness of the intersection under the weaker condition $\dim_{\mathrm{H}} S_{β,A} < 2$. This extends the previously known results on the real line. |
| title | Rational points in Cantor sets in the complex plane |
| topic | Number Theory Classical Analysis and ODEs 11A63, 28A80 |
| url | https://arxiv.org/abs/2512.07139 |