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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2308.03049 |
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Table of 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.