Representation and coding of rational pairs on a Triangular tree and\n Diophantine approximation in \ℝ²

2020/07/12 by Claudio Bonanno, Bonanno, Claudio, Alessio Del Vigna +1
Computer Science · Mathematics · #11B57 #37A44 #37E30 #Algorithms and Data Compression #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2007.05958

openalex publication_date 2020/07/12 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

In this paper we study the properties of the \Triangular tree, a\ncomplete tree of rational pairs introduced in citecas, in analogy with the\nmain properties of the Farey tree (or Stern-Brocot tree). To our knowledge the\nTriangular tree is the first generalisation of the Farey tree constructed using\nthe mediant operation. In particular we introduce a two-dimensional\nrepresentation for the pairs in the tree, a coding which describes how to reach\na pair by motions on the tree, and its description in terms of\nSL(3,\ℤ) matrices. The tree and the properties we study are then used\nto introduce rational approximations of non-rational pairs.\n

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