论文标题
在$ \ mathbb {r}^2 $中的三角形树和二芬太丁近似上的有理对的表示和编码
Representation and coding of rational pairs on a Triangular tree and Diophantine approximation in $\mathbb{R}^2$
论文作者
论文摘要
在本文中，我们研究了\ emph {三角树}的特性，这是\ cite {cas}中引入的完整的有理对树，类似于farey树（或stern-brocot树）的主要特性。据我们所知，三角树是使用Mediant操作构建的Farey树的第一个概括。特别是，我们为树上的对介绍了二维表示形式，该编码描述了如何通过树上的动作到达一对及其描述中的编码，并用$ SL（3，\ Mathbb {z}）$矩阵矩阵。然后，我们研究的树和我们研究的特性用于引入非理性对的合理近似值。
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.
