Constructing Displacement Vectors
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866929513000075264 |
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| author | Agin, Alon |
| author_facet | Agin, Alon |
| contents | Let $\overrightarrow{v}\in\mathbb{R}^2\setminus\mathbb{Q}^2$, let $\lVert\cdot\lVert$ be an arbitrary norm on $\mathbb{R}^2$, and let $(q_n,\overrightarrow{p_n})_{n=0}^{\infty} \subset\mathbb{N}\times\mathbb{Z}^{2}$ be the best approximation vectors sequence of $\overrightarrow{v}$ with respect to $\lVert\cdot\lVert$. We define the nth long displacement vector of $\overrightarrow{v}$ to be $\overrightarrow{β_n}:=\sqrt{q_{n+1}}(q_{n}\overrightarrow{v}-\overrightarrow{p_n})$ and prove the existence of long displacement vectors who have non-typical properties, focusing on their length, direction, and congruence class. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_03049 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Constructing Displacement Vectors Agin, Alon Number Theory 11J13 (Primary) 11H06 (Secondary) Let $\overrightarrow{v}\in\mathbb{R}^2\setminus\mathbb{Q}^2$, let $\lVert\cdot\lVert$ be an arbitrary norm on $\mathbb{R}^2$, and let $(q_n,\overrightarrow{p_n})_{n=0}^{\infty} \subset\mathbb{N}\times\mathbb{Z}^{2}$ be the best approximation vectors sequence of $\overrightarrow{v}$ with respect to $\lVert\cdot\lVert$. We define the nth long displacement vector of $\overrightarrow{v}$ to be $\overrightarrow{β_n}:=\sqrt{q_{n+1}}(q_{n}\overrightarrow{v}-\overrightarrow{p_n})$ and prove the existence of long displacement vectors who have non-typical properties, focusing on their length, direction, and congruence class. |
| title | Constructing Displacement Vectors |
| topic | Number Theory 11J13 (Primary) 11H06 (Secondary) |
| url | https://arxiv.org/abs/2308.03049 |