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

Abstract

In this paper we study the properties of the Triangular tree, a complete tree of rational pairs introduced in [7], 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, Z) matrices. The tree and the properties we study are then used to introduce rational approximations of non-rational pairs.

On integer partitions and continued fraction type algorithms

“…In the last step, exactly as in the construction of Section 4.1, we could have used F1 obtaining the partition (1, 1) × [7,12].…”

“…By now, multidimensional continued fractions provide a rich source of examples in dynamical systems, automata theory and many other areas. For background on and applications of the Triangle map for multidimensional continued fractions, see [4,12,16,19,21,6,11,7,5]. Set…”

“…For this reason we have chosen to make our algorithm to stop when reaching the segment x + y = 1. One of the possible extensions of T to the full boundary of △ has been studied in [6] and has been used in [7] to construct the Triangular tree. But going from △ to its boundary can be seen as a projection from a two-variable map to a one-variable map.…”

“…Key to our formula for p(2, n) and for the link between the Farey map and integer partitions of a positive number into two smaller numbers, with multiplicity, is the Farey tree and its underlying dynamics. Recently, two of the authors, with Sara Munday, [6] have developed an analogous tree structure for the slow-Triangle map (see also [7]). We plan on pursuing this approach in future work.…”