Representation and coding of rational pairs on a triangular tree and Diophantine approximation in $\mathbb R^2$
Volume 200 / 2021
Acta Arithmetica 200 (2021), 389-427
MSC: Primary 11B57; Secondary 37A44, 37E30.
DOI: 10.4064/aa200820-3-3
Published online: 11 October 2021
Abstract
We study the properties of the Triangular Tree, a complete tree of rational pairs introduced by Bonanno et al. (2021), 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 ${\rm SL}(3,\mathbb Z )$ matrices. The tree and the properties we study are then used to introduce rational approximations of non-rational pairs.
