Heights and transcendence of $p$--adic continued fractions
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909484317671424 |
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| author | Longhi, Ignazio Murru, Nadir Saettone, Francesco Maria |
| author_facet | Longhi, Ignazio Murru, Nadir Saettone, Francesco Maria |
| contents | Special kinds of continued fractions have been proved to converge to transcendental real numbers by means of the celebrated Subspace Theorem. In this paper we study the analogous $p$--adic problem. More specifically, we deal with Browkin $p$--adic continued fractions. First we give some new remarks about the Browkin algorithm in terms of a $p$--adic Euclidean algorithm. Then, we focus on the heights of some $p$--adic numbers having a periodic $p$--adic continued fraction expansion and we obtain some upper bounds. Finally, we exploit these results, together with $p$--adic Roth-like results, in order to prove the transcendence of two families of $p$--adic continued fractions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_04017 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Heights and transcendence of $p$--adic continued fractions Longhi, Ignazio Murru, Nadir Saettone, Francesco Maria Number Theory 11J70, 11J87 Special kinds of continued fractions have been proved to converge to transcendental real numbers by means of the celebrated Subspace Theorem. In this paper we study the analogous $p$--adic problem. More specifically, we deal with Browkin $p$--adic continued fractions. First we give some new remarks about the Browkin algorithm in terms of a $p$--adic Euclidean algorithm. Then, we focus on the heights of some $p$--adic numbers having a periodic $p$--adic continued fraction expansion and we obtain some upper bounds. Finally, we exploit these results, together with $p$--adic Roth-like results, in order to prove the transcendence of two families of $p$--adic continued fractions. |
| title | Heights and transcendence of $p$--adic continued fractions |
| topic | Number Theory 11J70, 11J87 |
| url | https://arxiv.org/abs/2302.04017 |