Constructing displacement vectors

Let $\vec{v} \in ℝ^{2} ∖ ℚ^{2}$ , let $∥ \cdot ∥$ be an arbitrary norm on $ℝ^{2}$ , and let ${(q_{n}, {\vec{p}}_{n})}_{n = 0}^{\infty} \subset ℕ \times ℤ^{2}$ be the best approximation vectors sequence of $\vec{v}$ with respect to $∥ \cdot ∥$ . We define the $n$ -th long displacement vector of $\vec{v}$ to be ${\vec{β}}_{n} : = \sqrt{q_{n + 1}} (q_{n} \vec{v} - {\vec{p}}_{n})$ and prove the existence of long displacement vectors who have nontypical properties — focusing on their length, direction, and congruence class.

PDF Access Denied

We have not been able to recognize your IP address 18.97.14.89 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

Keywords

displacement vector, best approximations, best approximation vectors, Dirichlet spectrum, Dirichlet converging, limiting Dirichlet spectrum

Mathematical Subject Classification

Primary: 11H06, 11J13

Milestones

Received: 13 August 2023

Revised: 22 May 2024

Accepted: 5 June 2024

Published: 23 September 2024

Authors

Alon Agin
School of Mathematical Sciences Tel Aviv University Tel Aviv Israel

Vol. 13, No. 3, 2024

Alon Agin

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors