If a machine did it, it is probably transcendental (even $p$-adically)
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909545834479616 |
|---|---|
| author | Capuano, Laura Checcoli, Sara Mula, Marzio Terracini, Lea |
| author_facet | Capuano, Laura Checcoli, Sara Mula, Marzio Terracini, Lea |
| contents | Continued fraction expansions provide a well-established bridge between algebraic properties of numbers and combinatorics on words. In this article, we investigate the algebraicity of $p$-adic numbers whose continued fractions arise from certain classes of words which generalize the classical automatic, periodic and palindromic words. Our main result shows that, under mild conditions on the $p$-adic continued fraction expansion, such numbers are either algebraic of degree at most 2 or transcendental. This result provides an analogue of results of Bugeaud and Adamczewski-Bugeaud in the real setting and extends previous works that were limited to specific choices of $p$-adic floor functions and less general classes of words. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16330 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | If a machine did it, it is probably transcendental (even $p$-adically) Capuano, Laura Checcoli, Sara Mula, Marzio Terracini, Lea Number Theory Combinatorics 11D88, 11J70, 11J81, 68R15 Continued fraction expansions provide a well-established bridge between algebraic properties of numbers and combinatorics on words. In this article, we investigate the algebraicity of $p$-adic numbers whose continued fractions arise from certain classes of words which generalize the classical automatic, periodic and palindromic words. Our main result shows that, under mild conditions on the $p$-adic continued fraction expansion, such numbers are either algebraic of degree at most 2 or transcendental. This result provides an analogue of results of Bugeaud and Adamczewski-Bugeaud in the real setting and extends previous works that were limited to specific choices of $p$-adic floor functions and less general classes of words. |
| title | If a machine did it, it is probably transcendental (even $p$-adically) |
| topic | Number Theory Combinatorics 11D88, 11J70, 11J81, 68R15 |
| url | https://arxiv.org/abs/2503.16330 |