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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| Date: | 2018 |
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| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2018
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/901 |
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| Journal Title: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Summary: | 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. |
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