Generalised Markov numbers

Abstract

In this paper we introduce generalised Markov numbers and extend the classical Markov theory for the discrete Markov spectrum to the case of generalised Markov numbers. In particular we show recursive properties for these numbers and find corresponding values in the Markov spectrum. Further we construct a counterexample to the generalised Markov uniqueness conjecture. The proposed generalisation is based on geometry of numbers. It substantively uses lattice trigonometry and geometric theory of continued numbers.

Introduction

In this paper we develop a geometric approach to the classical theory on the discrete Markov spectrum in terms of the geometric theory of continued fractions. We show that the principles hidden in the Markov's theory are much broader and can be substantively extended beyond the limits of Markov's theory. The aim of this paper is to introduce the generalisation of Markov theory and to make the first steps in its study.

Markov minima and Markov spectrum. Let us start with some classical definitions. Traditionally the Markov spectrum is defined via certain minima of binary quadratic forms (see [32], [33]). Let us define the Markov minima first.

Definition 1.1

Let f be a binary quadratic form with positive discriminant. The Markov minimum of f is

$$m(f)=\underset{{\mathbb{Z}}^2\setminus \{(0,0)\}}{\inf }|f|.$$

In some sense the Markov minimum gives us information on how far the locus of f is from the integer lattice (except for the origin). In this context it is reasonable to consider the following normalisation.

Definition 1.2

Let f be a binary quadratic form with positive discriminant $\mathrm{\Delta }(f)$. The normalised Markov minimum of f is

$$\mathcal{M}(f)=\frac{m(f)}{\sqrt{\mathrm{\Delta }(f)}}.$$

The set of all possible values for

$$\frac{1}{\mathcal{M}(f)}$$

is called the Markov Spectrum.

The functional $\mathcal{M}$ has various interesting properties. In particular if two forms $f_1$ and $f_2$ are proportional then their normalised Markov minima are the same. The lattice preserving linear transformations of the coordinates also do not change the value of the normalised Markov minima.

By that reason, $\mathcal{M}(f)$ depends only on the integer type (see Definition 2.4 below) of the arrangement of two lines forming the locus of f, which has an immediate trace in geometry of numbers.

Some history and background. The smallest element in the Markov spectrum is $\sqrt{5}$. It is defined by the form

$$x^2+xy-y^2$$

and therefore it is closely related to the golden ratio in geometry of numbers. The first elements in the Markov spectrum in increasing order are as follows:

$$\sqrt{5},\sqrt{8},\frac{\sqrt{221}}{5},\frac{\sqrt{1517}}{13},\frac{\sqrt{7565}}{29},\dots$$

The first two elements of the Markov spectrum were found in [29] by A. Korkine, G. Zolotareff. It turns out that the Markov spectrum is discrete at the segment $[\sqrt{5},3]$ except the element 3. This segment of Markov spectrum was studied by A. Markov in [32], [33]. We discuss his main results below.

The spectrum above the so-called Freiman's constant

$$F=\frac{2221564096+283748\sqrt{462)}}{491993569}=4.527829\dots$$

contains all real numbers (see [15]). The segment $[3,F]$ has a rather chaotic Markov spectrum, there are various open problems regarding it. There are numerous open gaps in this segment (i.e. open segments that do not contain any element of Markov spectrum). For the study of gaps we refer to a nice overview [8].

Remark 1.3

Note that most of the generalised Markov and almost Markov trees are entirely contained in the Markov spectrum above 3, which evidences the fractal nature of the spectrum.

The Markov spectrum has connections to different areas of mathematics, let us briefly mention some related references. Hyperbolic properties of Markov numbers were studied by C. Series in [36]. A. Sorrentino, K. Spalding, and A.P. Veselov have studied various properties of interesting monotone functions related to the Markov spectrum and the growth rate of values of binary quadratic forms in [38] and [37], [39]. B. Eren and A.M. Uludağ have described some properties of Jimm for certain Markov irrationals in [14]. In his paper [18] D. Gaifulin studied attainable numbers and the Lagrange spectrum (which is closely related to Markov spectrum).

Finally let us say a few words about the multidimensional case. One can consider a form of degree d in d variables corresponding to the product of d linear factors. The Markov minima and the d-dimensional Markov spectrum here are defined as in the two-dimensional case. It is believed that the d-dimensional Markov spectrum for $d>2$ is discrete, however this statement has not been proven yet. We refer the interested reader to the original manuscripts [9], [10], [11], [12], [13] by H. Davenport, and [41] by H.P.F. Swinnerton-Dyer, and to a nice overview in the book [20] by P.M. Gruber and C.G. Lekkerkerker.

Markov numbers and their properties. Let us recall an important and surprising theorem by A. Markov [32], [33] which relates the Markov Spectrum below 3 to certain binary quadratic forms and solutions to the Markov Diophantine equation

$$x^2+y^2+z^2=3xyz.$$

Definition 1.4

The solutions of this equation are called Markov triples. Elements of Markov triples are said to be Markov numbers.

Markov triples have a remarkable structure of a tree. This is due to the following three facts:

Fact 1

If $(a,b,c)$ is a solution to the Markov Diophantine equation then any permutation of $(a,b,c)$ is a solution as well.

Fact 2

If $(a,b,c)$ is a solution to the Markov Diophantine equation then the triple $(a,3ab-c,b)$ is a solution as well.

Fact 3

All possible compositions of the operations described in Fact 1, Fact 2 applied to the triple $(1,1,1)$ give rise to all positive solutions to Markov Diophantine equation.

Let us order the elements in triples $(a,b,c)$ as follows: $b\ge a\ge c$ and denote them by vertices. We also connect the vertex $(a,b,c)$ by a directed edge to the vertices $(a,3ab-c,b)$ and $(b,3bc-a,c)$ Then we have an arrangement of all the solutions as a graph which is actually a binary directed tree with the “long” root, see in Fig. 1.

The following famous Markov theorem links the triples of the Markov tree with the elements of the Markov spectrum below 3 by means of indefinite quadratic forms with integer coefficients.

Theorem 1.5

(A. Markov [33].) (i) The Markov spectrum below 3 consists of the numbers $\sqrt{9m^2-4}/m$, where m is a positive integer such that

$$m^2+m_1^2+m_2^2=3m{m}_1{m}_2,m_2\le m_1\le m,$$

for some positive integers $m_1$ and $m_2$.

(ii) Let the triple $(m,m_1,m_2)$ fulfil the conditions of item (i). Suppose that u is the least positive residue satisfying

$$m_{2}u\equiv \pm m_1\mathrm{mod}m$$

and v is defined from

$$u^2+1=vm.$$

Then the form

$$f_m(x,y)=m{x}^2+(3m-2u)xy+(v-3u)y^2$$

represents the value $\sqrt{9m^2-4}/m$ in the Markov spectrum.  □

The first steps to understand the phenomenon hidden in Markov's theorem were made by G. Frobenius [17] and R. Remak [35]. Following their works H. Cohn introduced special matrices in [4] and [5] (which were later called Cohn matrices) whose traces are three times Markov numbers (here H. Cohn used the trace identity of [16] by R. Fricke). As we show later (see Remark 4.3), the idea of Cohn matrices can be also extended to the case of generalised Markov and almost Markov trees, although the trace rule does not have a straightforward generalisation.

Main objectives of this paper. The generalisation of the classical Markov theory on the discrete Markov spectrum consists of the following major elements.

• First of all we introduce a geometric approach to the classical theory. (The outline see Diagram in Fig. 6.) This approach is based on interplay between continued fractions and convex geometry of lattice points in the cones.

• Basing on lattice geometry related to the classical case we construct generalised almost Markov and Markov triples of numbers (see Section 6.3).

• Further we relate generalised Markov triples to the elements of the Markov spectrum. We find Markov minima for the forms related to the generalised Markov trees in Corollary 7.11.

• Our next goal is to study the recursive properties of generalised Markov numbers (Corollary 7.19). These properties are essential for fast construction of the generalised Markov tree (the classical Markov tree is constructed iteratively by the formula of Fact 2 above).

• Finally we collect the main properties of the generalised Markov trees in Theorem 6.20 (see also the diagram in Fig. 8).

• We produce counterexamples for one of the generalised uniqueness conjectures, see Example 6.27, Example 6.28.

Organisation of the paper. We start in Section 2 with the definition of continued fractions and a discussion of lattice geometry techniques related to continued fractions.

Section 3 is dedicated to the classical Perron Identity. Our goal here is to relate the following objects: elements of the Markov spectrum, LLS sequences, reduced arrangements, and reduced forms.

Further in Section 4 we discuss the case of integer forms with integer coefficients. In this case all the corresponding LLS sequences are periodic. This property enriches the Perron Identity with additional interesting maps and relations. In particular the forms are linked now with extremal reduced $\mathrm{SL}(2,\mathbb{Z})$ matrices.

In Section 5 we introduce an important general triple-graph structure which perfectly fits to Markov theory and its generalisation proposed in this paper. After briefly defining triple-graphs in Subsection 5.1 we show several basic examples of triple-graph structure for Farey triples, Markov triples, triples of finite sequences, and triples of $\mathrm{SL}(2,\mathbb{Z})$-matrices (we refer to Subsections 5.2, 5.3, 5.4, and 5.5 respectively). Finally we study several important lexicographically monotone and algorithmic properties of these triple graphs, in Subsections 5.6 and 5.7.

We introduce the extended theory of Markov theory in Section 6. After a brief discussion of the classical case in Subsection 6.1 we formulate the main definitions and discuss the extended Markov theory in Subsections 6.2 and 6.3. We show the diagram for the extended Markov theory in Subsection 6.4 (see Theorem 6.20). Finally in Subsections 6.5 and 6.6 we discuss the uniqueness conjecture for both classical and generalised cases. In particular we show two counterexamples for one of the generalised triple graphs in Example 6.27, Example 6.28.

We conclude this article in Section 7 with proving all necessary statements used in the generalised Markov theorem.