Difference between revisions of "Markov spectrum problem"
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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|Binary quadratic form]]). Let | 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|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 | 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 | + | 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 | 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|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 | + | 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 | + | 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 | + | 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 | + | 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, | 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 | + | 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 | + | 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 < | + | For $ \lambda ( \theta ) < 3 $ |
+ | this problem reduces to the Markov problem; moreover, | ||
− | + | $$ | |
+ | L \cap [ 0 , 3 ) = M \cap [ 0 , 3 ) , | ||
+ | $$ | ||
− | and to each | + | 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 | + | 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 [[#References|[6]]]. For generalizations and analogues of the Markov spectrum problem and "isolation phenomena" see [[#References|[2]]], [[#References|[3]]], [[#References|[7]]]. | ||
====References==== | ====References==== | ||
Line 110: | Line 282: | ||
====Comments==== | ====Comments==== | ||
− | In equation (1) in the article above, the notation | + | 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 | + | 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 | + | 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==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D. Zagier, "On the number of Markoff numbers below a given bound" ''Math. Comp.'' , '''39''' (1982) pp. 709–723 {{MR|0669663}} {{ZBL|0501.10015}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> T.W. Cusick, M.E. Flahive, "The Markoff and Lagrange spectra" , Amer. Math. Soc. (1989) {{MR|1010419}} {{ZBL|0685.10023}} </TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D. Zagier, "On the number of Markoff numbers below a given bound" ''Math. Comp.'' , '''39''' (1982) pp. 709–723 {{MR|0669663}} {{ZBL|0501.10015}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> T.W. Cusick, M.E. Flahive, "The Markoff and Lagrange spectra" , Amer. Math. Soc. (1989) {{MR|1010419}} {{ZBL|0685.10023}} </TD></TR></table> |
Latest revision as of 07:59, 6 June 2020
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 |
Markov spectrum problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Markov_spectrum_problem&oldid=35323