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Constructing displacement vectors

Alon Agin

Vol. 13 (2024), No. 3, 239–264
Abstract

Let v 2 2, let be an arbitrary norm on 2, and let (qn,pn)n=0 × 2 be the best approximation vectors sequence of v with respect to . We define the n-th long displacement vector of v to be βn := qn+1(qnv pn) and prove the existence of long displacement vectors who have nontypical properties — focusing on their length, direction, and congruence class.

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