{"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/rational-matrix-digit-systems","title":"Rational matrix digit systems","arxiv_id":"2107.14168","date":"2021-07-29","proceeding":null,"authors":["Jonas Jankauskas","Jörg M. Thuswaldner"],"abstract":"Let $A$ be a $d \\times d$ matrix with rational entries which has no eigenvalue $\\lambda \\in \\mathbb{C}$ of absolute value $|\\lambda| < 1$ and let $\\mathbb{Z}^d[A]$ be the smallest nontrivial $A$-invariant $\\mathbb{Z}$-module. We lay down a theoretical framework for the construction of digit systems $(A, \\mathcal{D})$, where $\\mathcal{D}\\subset \\mathbb{Z}^d[A]$ finite, that admit finite expansions of the form \\[ \\mathbf{x}= \\mathbf{d}_0 + A \\mathbf{d}_1 + \\cdots + A^{\\ell-1}\\mathbf{d}_{\\ell-1} \\qquad(\\ell\\in \\mathbb{N},\\;\\mathbf{d}_0,\\ldots,\\mathbf{d}_{\\ell-1} \\in \\mathcal{D}) \\] for every element $\\mathbf{x}\\in \\mathbb{Z}^d[A]$. We put special emphasis on the explicit computation of small digit sets $\\mathcal{D}$ that admit this property for a given matrix $A$, using techniques from matrix theory, convex geometry, and the Smith Normal Form. Moreover, we provide a new proof of general results on this finiteness property and recover analogous finiteness results for digit systems in number fields a unified way.","url_abs":"https://arxiv.org/abs/2107.14168v2","url_pdf":"https://arxiv.org/pdf/2107.14168v2.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":"rational-matrix-digit-systems","repo_url":"https://github.com/jonas-jankauskas/DigitSystems","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"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}