Abstract

Let $\overrightarrow{v}\in {ℝ}^2\setminus {\mathbb{Q}}^2$, let $\parallel \cdot \parallel$ be an arbitrary norm on ${ℝ}^2$, 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 $\parallel \cdot \parallel$. We define the $n$-th long displacement vector of $\overrightarrow{v}$ to be ${\overrightarrow{\beta }}_n:=\sqrt{q_{n+1}}(q_n\overrightarrow{v}-{\overrightarrow{p}}_n)$ and prove the existence of long displacement vectors who have nontypical properties — focusing on their length, direction, and congruence class.

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