Markov spectrum problem

2020 Mathematics Subject Classification: Primary: 11J06 [MSN][ZBL]

A problem in number theory which arises in connection with the distribution of the normalized values of arithmetic minima of indefinite binary quadratic forms (cf. Binary quadratic form). Let

$$f = f ( x , y ) = \alpha x ^ {2} + \beta x y + \gamma y ^ {2} ,\ \ \alpha , \beta , \gamma \in \mathbf R ,$$

$$\delta ( f ) = \beta ^ {2} - 4 \alpha \gamma > 0 ,$$

and let

$$m ( f ) = \inf | f ( x , y ) | ,\ \ x , y \in \mathbf Z ^ {2} ,\ \ ( x , y ) \neq ( 0 , 0 ) ,$$

be the uniform arithmetic minimum of the form $f$. The number

$$\mu = \mu ( f ) = \ \frac{\sqrt {\delta ( f ) } }{m ( f ) } ,\ \ \mu \leq + \infty ,$$

is called the Markov constant of $f$. The set $M = \{ \mu ( f ) \}$, where $f$ runs through all real indefinite quadratic forms, is called the Markov spectrum. The Markov constant and the Markov spectrum have been defined in various ways; in particular, A.A. Markov in

considered the set $\{ 2 / \mu ( f ) \}$. It is known that $\mu ( f )$ is an invariant of a ray $F$ of classes of forms, that is, of a set

$$\tag{1 } F = \{ {f ^ { \prime } } : {f ^ { \prime } \simeq \tau f ( \mathbf Z ) ,\ \tau \in \mathbf R , \tau > 0 } \} ,$$

since $\mu ( f ^ { \prime } ) = \mu ( f ) = \mu ( F )$. Each ray of classes $F$ is in one-to-one correspondence with a doubly-infinite (infinite in both directions) sequence

$$I _ {F} = \{ {\dots, a _ {-} 1 ,\ a _ {0} , a _ {1} ,\dots } : {a _ {k} \in \mathbf Z } \} ,$$

such that if one puts

$$\mu _ {k} ( I _ {F} ) = \ [ a _ {k} ; a _ {k+} 1 , a _ {k+} 2 ,\dots ] + [ 0 ; a _ {k-} 1 , a _ {k-} 2 ,\dots ]$$

( $[ ; \dots ]$ is the notation for a continued fraction), then

$$\mu ( F ) = \sup _ {k \in \mathbf Z } \mu _ {k} ( I _ {F} ) .$$

The Markov problem can be stated as follows: 1) describe the Markov spectrum $M$; and 2) for each $\mu \in M$, describe the set of forms $f = f ( x , y)$( or the rays $F$) for which $\mu ( f ) = \mu ( F ) = \mu$. The problem was solved by Markov for the initial part of the spectrum $M$ defined by the condition $\mu ( f ) < 3$. This part of the spectrum is a discrete set:

$$M \cap [ 0 , 3 ) =$$

$$= \ \left \{ \sqrt {9 - \frac{4}{m ^ {2} } } : m ^ {2} + n ^ {2} + p ^ {2} = 3 m n p , m , n , p \in \mathbf N \right \} =$$

$$= \ \left \{ \sqrt 5 , \sqrt 8 , \frac{\sqrt 221 }{5} ,\dots \right \}$$

with the unique limit point 3 (a condensation point of $M$); $m$, $n$ and $p$ run through all positive integer solutions of Markov's Diophantine equation

$$\tag{2 } m ^ {2} + n ^ {2} + p ^ {2} = 3 m n p ,\ \ m \geq n \geq p > 0 .$$

In this case there corresponds to each point of this part of the spectrum precisely one ray $F _ {m}$, given by a Markov form $f _ {m} = f _ {m} ( x , y )$, with

$$\mu ( f _ {m} ) = \sqrt {9 - \frac{4}{m ^ {2} } } .$$

A solution $( m , n , p )$ of (2) is called a Markov triple; the number $m$ is called a Markov number. The Markov form $f _ {m}$ is associated to the Markov number $m = \max ( m , n , p )$ as follows. Let $r , s \in \mathbf Z$ be defined by the conditions

$$n r \equiv p ( \mathop{\rm mod} m ) ,\ \ 0 \leq r < m ,$$

$$r ^ {2} + 1 = m s ;$$

then, by definition,

$$f _ {m} = f _ {m} ( x , y ) = x ^ {2} + \left ( 3 - \frac{2r}{m} \right ) x y + \frac{s - 3 r }{m} y ^ {2} .$$

The set $M$ is closed and there is a smallest number $\mu _ {0} = 4.5278 \dots$ such that $[ \mu _ {0} , + \infty ] \subset M$ and $\mu _ {0}$ borders the interval of contiguity of $M$.

The Markov problem is closely related to the Lagrange–Hurwitz problem on rational approximation of a real number $\theta$. The quantity

$$\lambda = \lambda ( \theta ) = \sup \tau ,\ \ \lambda \leq + \infty ,$$

where the least upper bound is taken over all $\tau \in \mathbf R$, $\tau > 0$, for which

$$\left | \theta - \frac{p}{n} \right | \leq \frac{1}{\tau q ^ {2} }$$

has an infinite set of solutions $p , q \in \mathbf Z$, $q > 0$, is called a Lagrange constant. The set $L = \{ {\lambda ( \theta ) } : {\theta \in \mathbf R } \}$ is called the Lagrange spectrum. It is natural to regard Lagrange's theorem as the first result in the theory of the Lagrange spectrum: All convergents of the continued fraction expansion of $\theta$ satisfy

$$\left | \theta - \frac{p}{q} \right | < \frac{1}{q ^ {2} } .$$

If $\theta ^ \prime \sim \theta$, that is, if

$$\theta ^ \prime = \ \frac{a \theta + b }{c \theta + d } ,\ \ a , b , c , d \in \mathbf Z ,\ \ | a d - b c | = 1 ,$$

then $\lambda ( \theta ^ \prime ) = \lambda ( \theta ) = \lambda ( \Theta )$, where $\Theta = \{ {\theta ^ \prime } : {\theta ^ \prime \sim \theta } \}$ is an equivalence class of numbers. If $\theta$ is expanded as a continued fraction $\theta = [ a _ {0} ; a _ {1} , a _ {2} ,\dots ]$, then

$$\lambda ( \theta ) = \ \lim\limits _ {k \rightarrow \infty } \ \sup \lambda _ {k} ( \theta ) ,$$

$$\lambda _ {k} ( \theta ) = [ 0 ; a _ {k+} 1 , a _ {k+} 2 ,\dots ] + [ a _ {k} ; a _ {k-} 1 \dots a _ {1} ] ,$$

$$k = 1 , 2 ,\dots .$$

Thus, the Lagrange–Hurwitz problem can be stated as: a) describe the Lagrange spectrum $L$; and b) for each $\lambda \in L$, describe the set of numbers $\theta$( or classes $\Theta$) for which $\lambda ( \theta ) = \lambda ( \Theta ) = \lambda$.

For $\lambda ( \theta ) < 3$ this problem reduces to the Markov problem; moreover,

$$L \cap [ 0 , 3 ) = M \cap [ 0 , 3 ) ,$$

and to each $\lambda \in L$, $\lambda < 3$, corresponds precisely one class $\Theta$, described by the Markov form $f _ {m}$, provided the unicity conjecture is true. It has been proved that $L$, like $M$, is a closed set; that $L \subset M$ but $L \neq M$; that

$$L \cap [ \mu _ {0} , + \infty ] = \ M \cap [ \mu _ {0} , + \infty ] = \ [ \mu _ {0} , + \infty ] ,$$

where $\mu _ {0}$ borders the interval of contiguity of $L$. Research into the structure of $L$ and the connection between $L$ and $M$ is described in [6]. For generalizations and analogues of the Markov spectrum problem and "isolation phenomena" see [2], [3], [7].

References

[1a]	A. [A.A. Markov] Markoff, "Sur les formes quadratiques binaires indéfinies" Math. Ann. , 15 (1879) pp. 381–406 MR1510073 Zbl 11.0147.01
[1b]	A. [A.A. Markov] Markoff, "Sur les formes quadratiques binaires indéfinies" Math. Ann. , 17 (1880) pp. 379–400 MR1510073 Zbl 12.0143.02
[2]	J.W.S. Cassels, "An introduction to diophantine approximation" , Cambridge Univ. Press (1957) MR0087708 Zbl 0077.04801
[3]	B.N. Delone, "The Peterburg school of number theory" , Moscow-Leningrad (1947) (In Russian)
[4]	D.S. Gorshkov, "Lobachevskii geometry in connection with some problems of arithmetic" Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. , 67 (1977) pp. 39–85 (In Russian)
[5]	G.A. Freiman, "Diophantine approximation and the geometry of numbers. (The Markov problem)" , Kalinin (1975) (In Russian) Zbl 0347.10025
[6]	A.V. Malyshev, "Markov and Lagrange spectra (a survey of the literature)" Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. , 67 (1977) pp. 5–38 (In Russian)
[7]	B.A. Venkov, "On an extremum problem of Markov for indefinite ternaire quadratic forms" Izv. Akad. Nauk SSSR Ser. Mat. , 9 (1945) pp. 429–494 (In Russian) (French summary)

Comments

In equation (1) in the article above, the notation $f \simeq f ^ { \prime }$( $\mathbf Z$) refers to equivalence of binary forms over $\mathbf Z$. More precisely, $f \simeq f ^ { \prime }$( $\mathbf Z$) if and only if there are integers $a , b , c , d \in \mathbf Z$, $\mathop{\rm det} ( {} _ {c} ^ {a} {} _ {d} ^ {b} ) = \pm 1$ such that $f ^ { \prime } ( x , y ) = f ( a x + b y , c x + d y )$.

The "interval of contiguity of a Markov spectruminterval of contiguity" of $M$ is simply the maximal interval $[ \mu _ {0} , \infty ]$ completely belonging to $M$. The intersections $M \cap [ 0 , 3 )$ and $( \mu _ {0} , \infty ] \cap M$ have been well-described. The structure of the portion between, i.e. $M \cap [ 3 , \mu _ {0} ]$, is still (1989) unclear.

The unicity conjecture claims that the Markov number $m$ uniquely determines the triplet $( m,n,p,)$( and thus the Markov form $f _ {m}$). It is still (1989) a conjecture.

References

[a1]	D. Zagier, "On the number of Markoff numbers below a given bound" Math. Comp. , 39 (1982) pp. 709–723 MR0669663 Zbl 0501.10015
[a2]	T.W. Cusick, M.E. Flahive, "The Markoff and Lagrange spectra" , Amer. Math. Soc. (1989) MR1010419 Zbl 0685.10023