{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-the-arithmetic-of-multidimensional","title":"On the arithmetic of multidimensional continued fractions","arxiv_id":"2505.07810","date":"2025-05-12","proceeding":null,"authors":["Piotr Miska","Nadir Murru","Giuliano Romeo"],"abstract":"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 M\\\"obius 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.","url_abs":"https://arxiv.org/abs/2505.07810v1","url_pdf":"https://arxiv.org/pdf/2505.07810v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"on-the-arithmetic-of-multidimensional","repo_url":"https://github.com/giulianoromeont/continued-fractions","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}