Cobham's theorem for the Gaussian integers
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914179324051456 |
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| author | Bustos-Gajardo, Álvaro Fokkink, Robbert Yassawi, Reem |
| author_facet | Bustos-Gajardo, Álvaro Fokkink, Robbert Yassawi, Reem |
| contents | Assuming the four exponentials conjecture, Hansel and Safer showed that if a subset $S$ of the Gaussian integers is both $α=-m+i $- and $β=-n+i$-recognizable, then it is syndetic, and they conjectured that $S$ must be eventually periodic. Without assuming the four exponentials conjecture, we show that if $α$ and $β$ are multiplicatively independent Gaussian integers, and at least one of $α$, $β$ is not an $n$-th root of an integer, then any $α$- and $β$-automatic configuration is eventually periodic; in particular we prove Hansel and Safer's conjecture. Otherwise, there exist non-eventually periodic configurations which are $α$-automatic for any root of an integer $α$. Our work generalises the Cobham-Semenov theorem to Gaussian numerations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_01440 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cobham's theorem for the Gaussian integers Bustos-Gajardo, Álvaro Fokkink, Robbert Yassawi, Reem Number Theory Formal Languages and Automata Theory Commutative Algebra 11B85, 13F07, 68Q45 Assuming the four exponentials conjecture, Hansel and Safer showed that if a subset $S$ of the Gaussian integers is both $α=-m+i $- and $β=-n+i$-recognizable, then it is syndetic, and they conjectured that $S$ must be eventually periodic. Without assuming the four exponentials conjecture, we show that if $α$ and $β$ are multiplicatively independent Gaussian integers, and at least one of $α$, $β$ is not an $n$-th root of an integer, then any $α$- and $β$-automatic configuration is eventually periodic; in particular we prove Hansel and Safer's conjecture. Otherwise, there exist non-eventually periodic configurations which are $α$-automatic for any root of an integer $α$. Our work generalises the Cobham-Semenov theorem to Gaussian numerations. |
| title | Cobham's theorem for the Gaussian integers |
| topic | Number Theory Formal Languages and Automata Theory Commutative Algebra 11B85, 13F07, 68Q45 |
| url | https://arxiv.org/abs/2510.01440 |