Constructing Displacement Vectors

Let $\overrightarrow{v}\in {\mathbb{R}}^2\setminus {\mathbb{Q}}^2$, let $\parallel \cdot \parallel$ be an arbitrary norm on ${\mathbb{R}}^2$, and let $(q_n,\overrightarrow{p_n}{)}_{n=0}^{\mathrm{\infty }}\subset \mathbb{N}\times {\mathbb{Z}}^2$ be the best approximation vectors sequence of $\overrightarrow{v}$ with respect to $\parallel \cdot \parallel$. We define the nth 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 non-typical properties, focusing on their length, direction, and congruence class.