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