On the arithmetic of multidimensional continued fractions
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915284101627904 |
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| author | Miska, Piotr Murru, Nadir Romeo, Giuliano |
| author_facet | Miska, Piotr Murru, Nadir Romeo, Giuliano |
| contents | The problem of developing an arithmetic for continued fractions (in order to perform, e.g., sums and products) does not have a straightforward solution and has been addressed by several authors. In 1972, Gosper provided an algorithm to solve this problem. In this paper, we extend this approach in order to develop an arithmetic for multidimensional continued fractions (MCFs). First, we define the Möbius transform of an MCF and we provide an algorithm to obtain its expansion. Similarly, we deal with the bilinear transformation of MCFs, which covers as a special case the problem of summing or multiplying two MCFs. Finally, some experiments are performed in order to study the behavior of the algorithms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_07810 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the arithmetic of multidimensional continued fractions Miska, Piotr Murru, Nadir Romeo, Giuliano Number Theory 11J70, 11Y65 The problem of developing an arithmetic for continued fractions (in order to perform, e.g., sums and products) does not have a straightforward solution and has been addressed by several authors. In 1972, Gosper provided an algorithm to solve this problem. In this paper, we extend this approach in order to develop an arithmetic for multidimensional continued fractions (MCFs). First, we define the Möbius transform of an MCF and we provide an algorithm to obtain its expansion. Similarly, we deal with the bilinear transformation of MCFs, which covers as a special case the problem of summing or multiplying two MCFs. Finally, some experiments are performed in order to study the behavior of the algorithms. |
| title | On the arithmetic of multidimensional continued fractions |
| topic | Number Theory 11J70, 11Y65 |
| url | https://arxiv.org/abs/2505.07810 |