Real convergence and periodicity of $p$-adic continued fractions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908672194510848 |
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| author | Romeo, Giuliano |
| author_facet | Romeo, Giuliano |
| contents | Continued fractions have been generalized over the field of $p$-adic numbers, where it is still not known an analogue of the famous Lagrange's Theorem. In general, the periodicity of $p$-adic continued fractions is well studied and addressed as a hard problem. In this paper, we show a strong connection between periodic $p$--adic continued fractions and the convergence to real quadratic irrationals. In particular, in the first part we prove that the convergence in $\mathbb{R}$ is a necessary condition for the periodicity of the continued fractions of a quadratic irrational in $\mathbb{Q}_p$. Moreover, we leave several conjectures on the converse, supported by experimental computations. In the second part of the paper, we exploit these results to develop a probabilistic argument for the non-periodicity of Browkin's $p$-adic continued fractions. The probabilistic results are conditioned under the assumption of uniform distribution of the $p$-adic digits of a quadratic irrational, that holds for almost all $p$-adic numbers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_09215 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Real convergence and periodicity of $p$-adic continued fractions Romeo, Giuliano Number Theory 11J70, 11D88, 11Y65, 12J25 Continued fractions have been generalized over the field of $p$-adic numbers, where it is still not known an analogue of the famous Lagrange's Theorem. In general, the periodicity of $p$-adic continued fractions is well studied and addressed as a hard problem. In this paper, we show a strong connection between periodic $p$--adic continued fractions and the convergence to real quadratic irrationals. In particular, in the first part we prove that the convergence in $\mathbb{R}$ is a necessary condition for the periodicity of the continued fractions of a quadratic irrational in $\mathbb{Q}_p$. Moreover, we leave several conjectures on the converse, supported by experimental computations. In the second part of the paper, we exploit these results to develop a probabilistic argument for the non-periodicity of Browkin's $p$-adic continued fractions. The probabilistic results are conditioned under the assumption of uniform distribution of the $p$-adic digits of a quadratic irrational, that holds for almost all $p$-adic numbers. |
| title | Real convergence and periodicity of $p$-adic continued fractions |
| topic | Number Theory 11J70, 11D88, 11Y65, 12J25 |
| url | https://arxiv.org/abs/2410.09215 |