Representation and coding of rational pairs on a Triangular tree and Diophantine approximation in $\mathbb{R}^2$

TL;DR

This paper explores the properties of the Triangular tree, a novel structure generalizing the Farey tree to rational pairs in two dimensions, and applies it to Diophantine approximation in ^2.

Contribution

It introduces the Triangular tree, a new rational pair tree using mediant operations, along with a 2D representation, coding scheme, and matrix description, for Diophantine approximation.

Findings

The Triangular tree generalizes the Farey tree to rational pairs.

A new 2D representation and coding scheme for the tree are developed.

Applications to rational approximation of non-rational pairs are demonstrated.

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 $SL(3,\mathbb{Z})$ matrices. The tree and the properties we study are then used to introduce rational approximations of non-rational pairs.