Multivariable automatic arrays and transcendence
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917406657478656 |
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| author | Paul, Aadrita Ray, Anwesh |
| author_facet | Paul, Aadrita Ray, Anwesh |
| contents | We study real numbers defined by multidimensional automatic arrays weighted by multiplicatively independent bases. Let $a_1, \dots, a_r\geq 2$ be integers such that $\log a_1, \dots, \log a_r$ are $\mathbb Q$-linearly independent. Given bounded automatic sequences $(p_n(i))_{n\geq 0}$ with $i=1, \dots , r$ and a function $f:\mathbb Z^r\rightarrow \mathbb Z$, we consider the associated series $α= \sum_{n_1,\dots,n_r \geq 0} \frac{f(p_{n_1}(1),\dots,p_{n_r}(r))}{a_1^{n_1}\cdots a_r^{n_r}}$. Using combinatorial properties of automatic sequences and Schmidt's Subspace Theorem, we prove that $α$ is either rational or transcendental. This extends a result of Adamczewski and Bugeaud to the multidimensional setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_12468 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multivariable automatic arrays and transcendence Paul, Aadrita Ray, Anwesh Number Theory Combinatorics 11B85, 11Y16 (Primary), 11R04, 11A63 (Secondary) We study real numbers defined by multidimensional automatic arrays weighted by multiplicatively independent bases. Let $a_1, \dots, a_r\geq 2$ be integers such that $\log a_1, \dots, \log a_r$ are $\mathbb Q$-linearly independent. Given bounded automatic sequences $(p_n(i))_{n\geq 0}$ with $i=1, \dots , r$ and a function $f:\mathbb Z^r\rightarrow \mathbb Z$, we consider the associated series $α= \sum_{n_1,\dots,n_r \geq 0} \frac{f(p_{n_1}(1),\dots,p_{n_r}(r))}{a_1^{n_1}\cdots a_r^{n_r}}$. Using combinatorial properties of automatic sequences and Schmidt's Subspace Theorem, we prove that $α$ is either rational or transcendental. This extends a result of Adamczewski and Bugeaud to the multidimensional setting. |
| title | Multivariable automatic arrays and transcendence |
| topic | Number Theory Combinatorics 11B85, 11Y16 (Primary), 11R04, 11A63 (Secondary) |
| url | https://arxiv.org/abs/2604.12468 |