Representation and coding of rational pairs on a Triangular tree and Diophantine approximation in $\mathbb{R}^2$
Number Theory
2020-07-14 v1 Dynamical Systems
Abstract
In this paper we study the properties of the \emph{Triangular tree}, a complete tree of rational pairs introduced in \cite{cas}, in analogy with the main properties of the Farey tree (or Stern-Brocot tree). To our knowledge the Triangular tree is the first generalisation of the Farey tree constructed using the mediant operation. In particular we introduce a two-dimensional representation for the pairs in the tree, a coding which describes how to reach a pair by motions on the tree, and its description in terms of matrices. The tree and the properties we study are then used to introduce rational approximations of non-rational pairs.
Keywords
Cite
@article{arxiv.2007.05958,
title = {Representation and coding of rational pairs on a Triangular tree and Diophantine approximation in $\mathbb{R}^2$},
author = {Claudio Bonanno and Alessio Del Vigna},
journal= {arXiv preprint arXiv:2007.05958},
year = {2020}
}
Comments
30 pages, 8 figures