Gaussian rational numbers in Cantor sets in the complex plane
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909947145486336 |
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| author | Wu, Yu-Feng |
| author_facet | Wu, Yu-Feng |
| contents | Given $β\in\mathbb{Z}[i]$ with $|β|>1$ and a finite set $D\subset\mathbb{Q}(i)$, let \[K_{β, D}=\left\{\sum_{j=1}^{\infty}\frac{d_j}{β^j}: d_j\in D, \forall j\geq 1\right\}.\] Let $\mathcal{S}$ be a finite set of non-associate prime elements in $\mathbb{Z}[i]$ not dividing $β$. We prove that if the Hausdorff dimension of $K_{β,D}$ is less than $1$, then there are only finitely many Gaussian rational numbers in $K_{β,D}$ whose denominators have all their prime factors in $\mathcal{S}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_16281 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gaussian rational numbers in Cantor sets in the complex plane Wu, Yu-Feng Number Theory 11A63, 11J83, 28A80 Given $β\in\mathbb{Z}[i]$ with $|β|>1$ and a finite set $D\subset\mathbb{Q}(i)$, let \[K_{β, D}=\left\{\sum_{j=1}^{\infty}\frac{d_j}{β^j}: d_j\in D, \forall j\geq 1\right\}.\] Let $\mathcal{S}$ be a finite set of non-associate prime elements in $\mathbb{Z}[i]$ not dividing $β$. We prove that if the Hausdorff dimension of $K_{β,D}$ is less than $1$, then there are only finitely many Gaussian rational numbers in $K_{β,D}$ whose denominators have all their prime factors in $\mathcal{S}$. |
| title | Gaussian rational numbers in Cantor sets in the complex plane |
| topic | Number Theory 11A63, 11J83, 28A80 |
| url | https://arxiv.org/abs/2511.16281 |