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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m0625401.png" /></td> </tr></table>
+
$$
 +
= f ( x , y )  = \alpha x  ^ {2} + \beta x y + \gamma y  ^ {2} ,\ \
 +
\alpha , \beta , \gamma \in \mathbf R ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m0625402.png" /></td> </tr></table>
+
$$
 +
\delta ( f  )  = \beta  ^ {2} - 4 \alpha \gamma  > 0 ,
 +
$$
  
 
and let
 
and let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m0625403.png" /></td> </tr></table>
+
$$
 +
m ( f  )  = \inf  | f ( x , y ) | ,\ \
 +
x , y \in \mathbf Z  ^ {2} ,\ \
 +
( x , y ) \neq ( 0 , 0 ) ,
 +
$$
 +
 
 +
be the uniform arithmetic minimum of the form  $  f $.  
 +
The number
  
be the uniform arithmetic minimum of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m0625404.png" />. The number
+
$$
 +
\mu  = \mu ( f  )  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m0625405.png" /></td> </tr></table>
+
\frac{\sqrt {\delta ( f  ) } }{m ( f  ) }
 +
,\ \
 +
\mu \leq  + \infty ,
 +
$$
  
is called the Markov constant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m0625406.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m0625407.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m0625408.png" /> 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
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m0625409.png" />. It is known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254010.png" /> is an invariant of a ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254011.png" /> of classes of forms, that is, of a set
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
= \{ {f ^ { \prime } } : {f ^ { \prime } \simeq \tau f  ( \mathbf Z ) ,\
 +
\tau \in \mathbf R , \tau > 0 } \}
 +
,
 +
$$
  
since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254013.png" />. Each ray of classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254014.png" /> is in one-to-one correspondence with a doubly-infinite (infinite in both directions) sequence
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254015.png" /></td> </tr></table>
+
$$
 +
I _ {F}  = \{ {\dots, a _ {-} 1 ,\
 +
a _ {0} , a _ {1} ,\dots } : {a _ {k} \in \mathbf Z } \}
 +
,
 +
$$
  
 
such that if one puts
 
such that if one puts
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254016.png" /></td> </tr></table>
+
$$
 +
\mu _ {k} ( I _ {F} )  = \
 +
[ a _ {k} ; a _ {k+} 1 , a _ {k+} 2 ,\dots ] +
 +
[ 0 ; a _ {k-} 1 , a _ {k-} 2 ,\dots ]
 +
$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254017.png" /> is the notation for a [[Continued fraction|continued fraction]]), then
+
( $  [  ;  \dots ] $
 +
is the notation for a [[Continued fraction|continued fraction]]), then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254018.png" /></td> </tr></table>
+
$$
 +
\mu ( F  )  = \sup _ {k \in \mathbf Z }  \mu _ {k} ( I _ {F} ) .
 +
$$
  
The Markov problem can be stated as follows: 1) describe the Markov spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254019.png" />; and 2) for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254020.png" />, describe the set of forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254021.png" /> (or the rays <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254022.png" />) for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254023.png" />. The problem was solved by Markov for the initial part of the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254024.png" /> defined by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254025.png" />. This part of the spectrum is a discrete set:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254026.png" /></td> </tr></table>
+
$$
 +
M \cap [ 0 , 3 ) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254027.png" /></td> </tr></table>
+
$$
 +
= \
 +
\left \{ \sqrt {9 -  
 +
\frac{4}{m  ^ {2} }
 +
} : m  ^ {2} + n
 +
^ {2} + p  ^ {2} = 3 m n p , m , n , p \in \mathbf N \right \} =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254028.png" /></td> </tr></table>
+
$$
 +
= \
 +
\left \{ \sqrt 5 , \sqrt 8 ,
 +
\frac{\sqrt 221 }{5}
 +
,\dots \right \}
 +
$$
  
with the unique limit point 3 (a condensation point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254029.png" />); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254032.png" /> run through all positive integer solutions of Markov's Diophantine equation
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
m  ^ {2} + n  ^ {2} + p  ^ {2}  = 3 m n p ,\ \
 +
m \geq  n \geq  p > 0 .
 +
$$
  
In this case there corresponds to each point of this part of the spectrum precisely one ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254034.png" />, given by a Markov form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254035.png" />, with
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254036.png" /></td> </tr></table>
+
$$
 +
\mu ( f _ {m} )  = \sqrt {9 -  
 +
\frac{4}{m ^ {2} }
 +
} .
 +
$$
  
A solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254037.png" /> of (2) is called a Markov triple; the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254038.png" /> is called a Markov number. The Markov form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254039.png" /> is associated to the Markov number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254040.png" /> as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254041.png" /> be defined by the conditions
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254042.png" /></td> </tr></table>
+
$$
 +
n r  \equiv  p  (  \mathop{\rm mod}  m ) ,\ \
 +
0 \leq  r < m ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254043.png" /></td> </tr></table>
+
$$
 +
r  ^ {2} + 1  = m s ;
 +
$$
  
 
then, by definition,
 
then, by definition,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254044.png" /></td> </tr></table>
+
$$
 +
f _ {m}  = f _ {m} ( x , y )  = x  ^ {2} +
 +
\left ( 3 -
 +
\frac{2r}{m}
 +
\right ) x y +
 +
\frac{s - 3 r }{m}
 +
y  ^ {2} .
 +
$$
  
The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254045.png" /> is closed and there is a smallest number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254046.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254048.png" /> borders the interval of contiguity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254049.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254050.png" />. The quantity
+
The Markov problem is closely related to the Lagrange–Hurwitz problem on rational approximation of a real number $  \theta $.  
 +
The quantity
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254051.png" /></td> </tr></table>
+
$$
 +
\lambda  = \lambda ( \theta )  = \sup  \tau ,\ \
 +
\lambda \leq  + \infty ,
 +
$$
  
where the least upper bound is taken over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254053.png" />, for which
+
where the least upper bound is taken over all $  \tau \in \mathbf R $,
 +
$  \tau > 0 $,  
 +
for which
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254054.png" /></td> </tr></table>
+
$$
 +
\left | \theta -  
 +
\frac{p}{n}
 +
\right |  \leq 
 +
\frac{1}{\tau q  ^ {2} }
  
has an infinite set of solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254056.png" />, is called a Lagrange constant. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254057.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254058.png" /> satisfy
+
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254059.png" /></td> </tr></table>
+
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
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254060.png" />, that is, if
+
$$
 +
\left | \theta -
 +
\frac{p}{q}
 +
\right |  <
 +
\frac{1}{q  ^ {2} }
 +
.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254061.png" /></td> </tr></table>
+
If  $  \theta  ^  \prime  \sim \theta $,
 +
that is, if
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254062.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254063.png" /> is an equivalence class of numbers. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254064.png" /> is expanded as a continued fraction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254065.png" />, then
+
$$
 +
\theta  ^  \prime  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254066.png" /></td> </tr></table>
+
\frac{a \theta + b }{c \theta + d }
 +
,\ \
 +
a , b , c , d \in \mathbf Z ,\ \
 +
| a d - b c | = 1 ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254067.png" /></td> </tr></table>
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254068.png" /></td> </tr></table>
+
$$
 +
\lambda ( \theta )  = \
 +
\lim\limits _ {k \rightarrow \infty } \
 +
\sup  \lambda _ {k} ( \theta ) ,
 +
$$
  
Thus, the Lagrange–Hurwitz problem can be stated as: a) describe the Lagrange spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254069.png" />; and b) for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254070.png" />, describe the set of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254071.png" /> (or classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254072.png" />) for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254073.png" />.
+
$$
 +
\lambda _ {k} ( \theta ) = [ 0 ; a _ {k+} 1 , a _ {k+} 2
 +
,\dots ] + [ a _ {k} ;  a _ {k-} 1 \dots a _ {1} ] ,
 +
$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254074.png" /> this problem reduces to the Markov problem; moreover,
+
$$
 +
= 1 , 2 ,\dots .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254075.png" /></td> </tr></table>
+
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 $.
  
and to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254077.png" />, corresponds precisely one class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254078.png" />, described by the Markov form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254079.png" />, provided the unicity conjecture is true. It has been proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254080.png" />, like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254081.png" />, is a closed set; that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254082.png" /> but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254083.png" />; that
+
For  $  \lambda ( \theta ) < 3 $
 +
this problem reduces to the Markov problem; moreover,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254084.png" /></td> </tr></table>
+
$$
 +
L \cap [ 0 , 3 )  = M \cap [ 0 , 3 ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254085.png" /> borders the interval of contiguity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254086.png" />. Research into the structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254087.png" /> and the connection between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254089.png" /> is described in [[#References|[6]]]. For generalizations and analogues of the Markov spectrum problem and "isolation phenomena" see [[#References|[2]]], [[#References|[3]]], [[#References|[7]]].
+
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
  
====References====
+
$$
<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> A. [A.A. Markov] Markoff, "Sur les formes quadratiques binaires indéfinies" ''Math. Ann.'' , '''15''' (1879) pp. 381–406 {{MR|1510073}} {{ZBL|11.0147.01}} </TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> A. [A.A. Markov] Markoff, "Sur les formes quadratiques binaires indéfinies" ''Math. Ann.'' , '''17''' (1880) pp. 379–400 {{MR|1510073}} {{ZBL|12.0143.02}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.W.S. Cassels, "An introduction to diophantine approximation" , Cambridge Univ. Press (1957) {{MR|0087708}} {{ZBL|0077.04801}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> B.N. Delone, "The Peterburg school of number theory" , Moscow-Leningrad (1947) (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> D.S. Gorshkov, "Lobachevskii geometry in connection with some problems of arithmetic" ''Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov.'' , '''67''' (1977) pp. 39–85 (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> G.A. Freiman, "Diophantine approximation and the geometry of numbers. (The Markov problem)" , Kalinin (1975) (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> A.V. Malyshev, "Markov and Lagrange spectra (a survey of the literature)" ''Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov.'' , '''67''' (1977) pp. 5–38 (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> B.A. Venkov, "On an extremum problem of Markov for indefinite ternaire quadratic forms" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''9''' (1945) pp. 429–494 (In Russian) (French summary)</TD></TR></table>
+
L \cap [ \mu _ {0} , + \infty ] = \
 +
M \cap [ \mu _ {0} , + \infty ]  = \
 +
[ \mu _ {0} , + \infty ] ,
 +
$$
  
 +
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 [[#References|[6]]]. For generalizations and analogues of the Markov spectrum problem and "isolation phenomena" see [[#References|[2]]], [[#References|[3]]], [[#References|[7]]].
  
 +
====References====
 +
<table>
 +
<TR><TD valign="top">[1a]</TD> <TD valign="top"> A. [A.A. Markov] Markoff, "Sur les formes quadratiques binaires indéfinies" ''Math. Ann.'' , '''15''' (1879) pp. 381–406 {{MR|1510073}} {{ZBL|11.0147.01}} </TD></TR>
 +
<TR><TD valign="top">[1b]</TD> <TD valign="top"> A. [A.A. Markov] Markoff, "Sur les formes quadratiques binaires indéfinies" ''Math. Ann.'' , '''17''' (1880) pp. 379–400 {{MR|1510073}} {{ZBL|12.0143.02}} </TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top"> J.W.S. Cassels, "An introduction to diophantine approximation" , Cambridge Univ. Press (1957) {{MR|0087708}} {{ZBL|0077.04801}} </TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top"> B.N. Delone, "The Peterburg school of number theory" , Moscow-Leningrad (1947) (In Russian)</TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top"> 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)</TD></TR>
 +
<TR><TD valign="top">[5]</TD> <TD valign="top"> G.A. Freiman, "Diophantine approximation and the geometry of numbers. (The Markov problem)" , Kalinin (1975) (In Russian) {{ZBL|0347.10025}}</TD></TR>
 +
<TR><TD valign="top">[6]</TD> <TD valign="top"> 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)</TD></TR>
 +
<TR><TD valign="top">[7]</TD> <TD valign="top"> 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)</TD></TR>
 +
</table>
  
 
====Comments====
 
====Comments====
In equation (1) in the article above, the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254090.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254091.png" />) refers to equivalence of binary forms over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254092.png" />. More precisely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254093.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254094.png" />) if and only if there are integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254096.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254097.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254098.png" /> is simply the maximal interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m06254099.png" /> completely belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m062540100.png" />. The intersections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m062540101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m062540102.png" /> have been well-described. The structure of the portion between, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m062540103.png" />, is still (1989) unclear.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m062540104.png" /> uniquely determines the triplet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m062540105.png" /> (and thus the Markov form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m062/m062540/m062540106.png" />). It is still (1989) a conjecture.
+
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
How to Cite This Entry:
Markov spectrum problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Markov_spectrum_problem&oldid=33786
This article was adapted from an original article by A.V. Malyshev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article