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

The theory of multidimensional continued fractions has received increasing attention in the last years from researchers both in number theory and in ergodic theory. The origin of multidimensional continued fraction expansions may be traced back to a letter that Hermite sent to Jacobi asking for a generalisation of Lagrange’s Theorem for quadratic irrationals to algebraic irrationals of higher degree. It was for this reason that Jacobi developed what is now called the Jacobi-Perron algorithm. Unfortunately, despite numerous attempts and the introduction of many different algorithms, Hermite’s question remains unanswered. We refer the reader to [5] for a geometric description of the theory of multidimensional continued fractions. On the other hand, it is well-known that (regular) continued fraction expansions are related to the theory of dynamical systems as the expansion of a real number can be obtained by the symbolic representation of its orbit under the action of the Gauss map on the unit interval. The dynamical systems approach has led to new proofs of the Gauss-Kuzmin Theorem, Khinchin’s weak law and other metric results first obtained by Khinchin and Lévy, also thanks to the modern results of ergodic theory (see for example [11, 13]). In recent years the methods of ergodic theory have been applied also to maps related to multidimensional continued fraction algorithms, we refer to [15] for the first results in this research area. It is also interesting to mention that, in the opposite direction, number theoretical properties of real numbers have led to new results in ergodic theory, as the Three Gaps Theorem for instance, which can be interpreted in terms of return times for irrational rotations on the circle.

In this paper we continue the work initiated by the authors with Sara Munday in [7], where we have studied the properties of a tree of rational pairs, here called the Triangular tree, which was introduced as a two-dimensional version of the well-known Farey tree (or Stern-Brocot tree). The aim of this paper is to show that all the structures of the Farey tree can be found also in the Triangular tree and to construct approximations of real pairs using the tree. We believe that the Triangular tree will be useful for the study of interesting phenomena related to two-parameter systems, as this is the case for the Farey tree and one-parameter systems. For recent investigations on higher-dimensional phenomena we refer to [3, 4, 10].

The paper is structured as follows. In Section 2 we recall the construction of the Farey tree and its main properties based on the continued fraction expansions of real numbers, in particular the $\{L,R\}$ coding of real numbers which is based on the relations of the Farey tree with the group $SL(2,\mathbb{Z})$. In Section 3 we recall the main results of [7] concerning the Triangular tree. In particular we recall that it can be constructed in a dynamical way, by using the Triangle map defined in [8] which acts on the triangle $\triangle\coloneqq\{(x,y)\in\mathbb{R}^{2}:1\geq x\geq y>0\}$ and its slow version $S$ introduced in [7], and in a geometric way, by using the notion of mediant of two pairs of fractions. The existence of these two possible constructions is the first basic property of the Farey tree that has been proved for the Triangular tree in [7]. Afterwards, Section 4 contains some preparatory results on the triangle sequences, that is the symbolic representation of the orbits of real pairs for the Triangle map. These sequences are the analogous of the continued fraction expansions for real numbers and the Gauss map, and are thus the two-dimensional continued fraction expansions of real pairs generated by the Triangle map. It is known that there are cases in which different real pairs have the same triangle sequence, this is discussed in details in Section 4.

Sections 5 and 6 contain the main results of the paper. First we use the slow map $S$ and its local inverses to introduce in Definition 5.3 a unique two-dimensional representation for rational pairs in $\widebar{\triangle}$, thus improving the expansions obtained by using the Triangle map. For non-rational pairs the two-dimensional representation is that obtained by the Triangle map. Then in Theorem 6.6 we introduce a coding for rational pairs in the Triangular tree in terms of possible motions in the tree, in analogy with the coding of the Farey tree. This coding can be described also in terms of $SL(3,\mathbb{Z})$ matrices defined in (6.1). This coding and its descriptions are useful also to introduce approximations of real pairs, and of non-rational pairs in particular, in terms of the rational pairs on the Triangular tree. The definition of the approximations together with some examples are described in Section 7. Finally, in Section 8 we study the speed of the approximations introduced in the sense of the simultaneous approximations of couples of real numbers. We give results only for two classes of non-rational pairs, those with finite triangle sequence and those corresponding to fixed points of the Triangle map, leaving further developments for future research.

In this section we recall the construction of the Farey tree, the coding it induces for the rationals in $(0,1)$, and its connection with the continued fraction expansion of real numbers.

The Farey tree is a binary tree which contains all the rational numbers in the interval $(0,1)$ and can be generated in a dynamical way using the Farey map, and in an arithmetic way using the notion of mediant between two fractions.

The Farey map is the map $F:[0,1]\rightarrow[0,1]$ defined to be

Denoting with $F_{0}:[0,\frac{1}{2}]\rightarrow[0,1]$ and $F_{1}:[\frac{1}{2},1]\rightarrow[0,1]$ its two branches, $F$ admits two local inverses, $\psi_{0}\coloneqq F_{0}^{-1}$ and $\psi_{1}\coloneqq F_{1}^{-1}$, given by

The Farey tree is then generated using the Farey map by setting $\mathcal{L}_{0}\coloneqq\{\frac{1}{2}\}$ for the root of the tree, and setting recursively $\mathcal{L}_{n}\coloneqq F^{-n}\left(\frac{1}{2}\right)$ for all $n\geq 1$. Each level of the tree is written by ascending order as shown in Figure 1. The connections between the fractions of different levels are explained below. It is known that the Farey tree contains all the rational numbers in the interval $(0,1)$, that is $\bigcup_{n=0}^{+\infty}\mathcal{L}_{n}=\bigcup_{n=0}^{+\infty}F^{-n}\left(\frac{1}{2}\right)=\mathbb{Q}\cap(0,1)$, and each rational number appears in the tree exactly once.

We now describe the second way to construct the levels of the Farey tree. Let us consider the Stern-Brocot sets $(\mathcal{F}_{n})_{n\geq-1}$, with $\mathcal{F}_{-1}\coloneqq\left\{\frac{0}{1},\frac{1}{1}\right\}$, and for all $n\geq 0$ let $\mathcal{F}_{n}$ be obtained from $\mathcal{F}_{n-1}$ by inserting the mediant of each pair of neighbouring fractions. We recall that the mediant of two fractions $\frac{p}{q}$ and $\frac{r}{s}$ with $\frac{p}{q}<\frac{r}{s}$ is

and that $\frac{p}{q}<\frac{p}{q}\oplus\frac{r}{s}<\frac{r}{s}$. We say that $\frac{p}{q}\oplus\frac{r}{s}$ is the child of $\frac{p}{q}$ and $\frac{r}{s}$, which are its left and right parent, respectively. It can be easily shown that two fractions $\frac{p}{q}<\frac{r}{s}$ are neighbours in a Stern-Brocot set if and only if $qr-ps=1$ (see, for instance, [9]). Moreover, since the ancestors $\frac{0}{1}$ and $\frac{1}{1}$ are in lowest terms, it follows that all the fractions obtained through the mediant operation appear in lowest terms. The first sets $\mathcal{F}_{n}$ are

The levels of the Farey tree are then given by $\mathcal{L}_{n}=\mathcal{F}_{n}\setminus\mathcal{F}_{n-1}$ for all $n\geq 0$. Two fractions in Figure 1 are connected if one of the fractions is a parent of the other. Notice that a fraction in a level $\mathcal{L}_{n}$ has a parent in the level $\mathcal{L}_{n-1}$ and the other parent in a level $\mathcal{L}_{m}$ with $m<n-1$.

For later use we recall the notion of rank of a rational number $x$ in the open unit interval: $\operatorname*{rank}(x)=r$ if and only if $x\in\mathcal{L}_{r}$. For more details on the Farey tree, we refer to [6].

It is known that a triangle sequence does not necessarily represent a unique pair of real numbers, but could correspond to an entire line segment. If the triangle sequence terminates, we do not have uniqueness and Lemma 5.1 characterises the points having a given finite triangle sequence. Uniqueness is not guaranteed even when the triangle sequence is infinite: [8] gives a sufficient condition to have uniqueness and a criterion, equivalent to uniqueness is proved in the later work [1]. In this section we discuss this problem.

We start by introducing some notation. Let $X=\begin{pmatrix}q\\ p\\ r\end{pmatrix}$ be a three-dimensional vector with integer components and $q\neq 0$. We then define the correspondent rational pair

for which both components have the same denominator. For instance, the vertices $v_{0}$, $v_{1}$ and $v_{2}$ of $\triangle$ are represented by $\begin{pmatrix}1\\ 0\\ 0\end{pmatrix}$, $\begin{pmatrix}1\\ 1\\ 0\end{pmatrix}$, and $\begin{pmatrix}1\\ 1\\ 1\end{pmatrix}$, respectively. Note that the sum of two three-dimensional vectors corresponds to the mediant between the two correspondent two-dimensional vectors, that is

For a sequence $(\alpha_{0},\alpha_{1},\ldots)$ of non-negative integers and for an integer $k\geq 0$ we define

The set $\triangle(\alpha_{0},\ldots,\alpha_{k})$ is a triangle and consists of all those points whose first $k+1$ triangle sequence digits are precisely $\alpha_{0},\ldots,\alpha_{k}$ and thus these triangles are nested, that is

Let $(X_{k})_{k\geq-3}$ be the sequence of three-dimensional vectors defined as follows:

then the vertices of $\triangle(\alpha_{0},\ldots,\alpha_{k})$ are $\hat{X}_{k-1}$, $\hat{X}_{k}$ and $\hat{X}_{k-2}\oplus\hat{X}_{k}$ (see [1, Theorem 3]). Figure 5 shows the recursive construction of the triangles $\triangle(\alpha_{0},\ldots,\alpha_{k})$.

When the triangle sequence is infinite, the infinite intersection $\bigcap_{k\geq 0}\triangle(\alpha_{0},\ldots,\alpha_{k})$ can be either a point or a line segment.

1. (1)

   In the first case the nested triangles shrink to a point, which means that the triangle sequence $(\alpha_{k})_{k\geq 0}$ denotes a unique pair of real numbers $(\alpha,\beta)$. As a consequence $\operatorname*{diam}\triangle(\alpha_{0},\ldots,\alpha_{k})\rightarrow 0$ and the sequence $(\hat{X}_{k})_{k\geq-3}$ converges to $(\alpha,\beta)$. We will refer to this case as the convergent case.

2. (2)

   In the second case the triangle sequence does not uniquely describe a point but instead identifies a line segment $\mathfrak{L}$ of length $l>0$, such that all the points of $\mathfrak{L}$ have the same triangle sequence $[\alpha_{0},\alpha_{1},\ldots]$. In this case $\operatorname*{diam}\triangle(\alpha_{0},\ldots,\alpha_{k})\rightarrow l$ and the sequence $(\hat{X}_{k})_{k\geq-3}$ does not admit a limit. More precisely, we have that the odd and even terms of $(\hat{X}_{k})_{k\geq-3}$ converge to the two endpoints of $\mathfrak{L}$ [1, Theorem 6]. In particular, $d(\hat{X}_{k-1},\hat{X}_{k})\rightarrow l$ and it also holds that $d(\hat{X}_{k},\hat{X}_{k-1}\oplus\hat{X}_{k})\rightarrow 0$, where $d(\cdot,\cdot)$ is the Euclidean distance in $\mathbb{R}^{2}$. We will refer to this case as the non-convergent case.

The main result of [1, Section 6] is a criterion of uniqueness, which we now state. Let

and refer again to Figure 5 for the geometric interpretation of this quantity. The triangle sequence $[\alpha_{0},\alpha_{1},\ldots]$ does not correspond to a unique pair of real numbers if and only if it contains only a finite number of zeroes and $\prod_{k\geq N}(1-\lambda_{k})>0$, where $N$ is such that $\alpha_{k}>0$ for all $k\geq N$²²2Note that $\lambda_{k}=1$ if and only if $\alpha_{k}=0$.. The convergence of the infinite product to a non-zero number is equivalent to the convergence of $\sum_{k\geq 0}\lambda_{k}$, which is in turn equivalent to $\lambda_{k}\rightarrow 0$ sufficiently fast: the geometric meaning of $\lambda_{k}$ suggests that this condition is equivalent to a sufficiently fast growth of the triangle sequence digits $\alpha_{k}$.

We now prove some results of convergence for the points of the closed triangle $\widebar{\triangle}$ in the non-convergent case. For two rational pairs $\hat{X}$ and $\hat{Y}$ and a non-negative integer $s$ we introduce the notation

and recall that the maps $\phi_{0}$ and $\phi_{1}$ commute with the mediant operation for rational pairs whose components have the same denominator, that is

Also, in the following we denote by $\phi_{0}$ and $\phi_{1}$ the continuous extension to $\widebar{\triangle}$ of the maps defined in (3.3) and (3.4).