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arXiv · 2007.05958

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

Claudio Bonanno·Alessio Del Vigna
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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.

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Claudio Bonanno, Alessio Del Vigna. 2020-07-12. Representation and coding of rational pairs on a Triangular tree and Diophantine approximation in $\mathbb{R}^2$. https://arxiv.org/abs/2007.05958

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