Arithmetic properties of exceptional lattice paths
For a fixed real number \(\rho>0\), let \(L\) be an affine line of slope \(\rho^{-1}\) in \(\mathbb{R}^2\). We show that the closest approximation of \(L\) by a path \(P\) in \(\mathbb{Z}^2\) is unique, except in one case, up to integral translation. We study this exceptional case. For irrat...
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| Datum: | 2018 |
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| Sprache: | Englisch |
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Lugansk National Taras Shevchenko University
2018
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| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/901 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543323273035776 |
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| author | Rump, Wolfgang |
| author_facet | Rump, Wolfgang |
| author_sort | Rump, Wolfgang |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-03-21T07:07:28Z |
| description | For a fixed real number \(\rho>0\), let \(L\) be an affine line of slope \(\rho^{-1}\) in \(\mathbb{R}^2\). We show that the closest approximation of \(L\) by a path \(P\) in \(\mathbb{Z}^2\) is unique, except in one case, up to integral translation. We study this exceptional case. For irrational \(\rho\), the projection of \(P\) to \(L\) yields two quasicrystallographic tilings in the sense of Lunnon and Pleasants [5]. If \(\rho\) satisfies an equation \(x^2=mx+1\) with \(m\in\mathbb{Z}\), both quasicrystals are mapped to each other by a substitution rule. For rational \(\rho\), we characterize the periodic parts of \(P\) by geometric and arithmetic properties, and exhibit a relationship to the hereditary algebras \(H_{\rho}(K)\) over a field \(K\) introduced in a recent proof of a conjecture of Roiter. |
| first_indexed | 2025-12-02T15:28:35Z |
| format | Article |
| id | admjournalluguniveduua-article-901 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:28:35Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-9012018-03-21T07:07:28Z Arithmetic properties of exceptional lattice paths Rump, Wolfgang Lattice path, uniform enumeration, quasicrystal 05B30, 11B50; 52C35, 11A07 For a fixed real number \(\rho>0\), let \(L\) be an affine line of slope \(\rho^{-1}\) in \(\mathbb{R}^2\). We show that the closest approximation of \(L\) by a path \(P\) in \(\mathbb{Z}^2\) is unique, except in one case, up to integral translation. We study this exceptional case. For irrational \(\rho\), the projection of \(P\) to \(L\) yields two quasicrystallographic tilings in the sense of Lunnon and Pleasants [5]. If \(\rho\) satisfies an equation \(x^2=mx+1\) with \(m\in\mathbb{Z}\), both quasicrystals are mapped to each other by a substitution rule. For rational \(\rho\), we characterize the periodic parts of \(P\) by geometric and arithmetic properties, and exhibit a relationship to the hereditary algebras \(H_{\rho}(K)\) over a field \(K\) introduced in a recent proof of a conjecture of Roiter. Lugansk National Taras Shevchenko University 2018-03-21 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/901 Algebra and Discrete Mathematics; Vol 5, No 3 (2006) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/901/430 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | Lattice path uniform enumeration quasicrystal 05B30 11B50; 52C35 11A07 Rump, Wolfgang Arithmetic properties of exceptional lattice paths |
| title | Arithmetic properties of exceptional lattice paths |
| title_full | Arithmetic properties of exceptional lattice paths |
| title_fullStr | Arithmetic properties of exceptional lattice paths |
| title_full_unstemmed | Arithmetic properties of exceptional lattice paths |
| title_short | Arithmetic properties of exceptional lattice paths |
| title_sort | arithmetic properties of exceptional lattice paths |
| topic | Lattice path uniform enumeration quasicrystal 05B30 11B50; 52C35 11A07 |
| topic_facet | Lattice path uniform enumeration quasicrystal 05B30 11B50; 52C35 11A07 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/901 |
| work_keys_str_mv | AT rumpwolfgang arithmeticpropertiesofexceptionallatticepaths |