Constructing Displacement Vectors

TL;DR

This paper investigates the properties of long displacement vectors derived from best approximation sequences of irrational vectors in two dimensions, revealing the existence of vectors with atypical length, direction, and congruence characteristics.

Contribution

It introduces the concept of long displacement vectors and proves the existence of such vectors with non-typical properties, expanding understanding of approximation behavior.

Findings

Existence of long displacement vectors with non-typical properties.

Characterization of length, direction, and congruence class of these vectors.

Insights into approximation sequences of irrational vectors.

Abstract

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 $\overrightarrow{\beta_n}:=\sqrt{q_{n+1}}(q_{n}\overrightarrow{v}-\overrightarrow{p_n})$ and prove the existence of long displacement vectors who have non-typical properties, focusing on their length, direction, and congruence class.