5
$\begingroup$

We use the notation

$$\displaystyle [a_0; a_1, \cdots, a_n] = a_0 + \cfrac{1}{a_1 + \cfrac{1}{\ddots + \cfrac{1}{a_n}}}$$

to denote a finite continued fraction, and for a given real number $\alpha$, we attach the (possibly infinite) sequence of partial quotients $[a_0; a_1, a_2, \cdots] = \alpha$.

It is well-known by classic theorems of Euler and Lagrange that a real number has eventually periodic continued fraction expansion if and only if $\alpha$ is a quadratic irrational. Moreover, it is known that if $\alpha$ is an irrational number and $p/q$ is a reduced fraction with $q > 0$ such that $|\alpha - p/q| < 1/(2q^2)$, then $p/q$ is in fact a convergent of $\alpha$; i.e., there is a positive integer $k$ such that $p/q = [a_0; a_1, \cdots, a_k]$.

Suppose that $\theta$ is a purely periodic quadratic irrational, so that its continued fraction is of the form $[\overline{a_0; a_1, \cdots, a_n}]$. Let $f(x) = ax^2 + bx + c$ be the minimal polynomial of $\theta$. What do we know about the evaluation of the quadratic form $F(x,y) = y^2 f(x/y)$ at the pairs $(p_0, q_0), \cdots, (p_n, q_n)$ corresponding to the convergents? In particular, what are the sizes of these values compared to the discriminant $\Delta(F)$ of $F$?

$\endgroup$
4
$\begingroup$

The item you want is the neighbor method, a version of continued fractions. I learned this from Buell, Binary Quadratic Forms. It is also in a 1929 Introduction by Dickson, and a book by Matthews I've never seen. For a positive but not square discriminant, all primitively represented numbers with absolute value below $\frac{1}{2} \sqrt \Delta$ appear as coefficients (first or third) in the chain. There are usually a few more coefficients with slightly larger absolute values. Each triple in a cycle is a "reduced" indefinite form. A form $ax^2 + b xy + c y^2$ is reduced if and only if $ac<0$ and $b > |a+c| \; .$

enter image description here

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle 1 3 -2

  0  form              1           3          -2


           1           0
           0           1

To Return  
           1           0
           0           1

0  form   1 3 -2   delta  -1     ambiguous  
1  form   -2 1 2   delta  1
2  form   2 3 -1   delta  -3
3  form   -1 3 2   delta  1     ambiguous            -1 composed with form zero  
4  form   2 1 -2   delta  -1
5  form   -2 3 1   delta  3
6  form   1 3 -2


  form   1 x^2  + 3 x y  -2 y^2 

minimum was   1rep   x = 1   y = 0 disc 17 dSqrt 4  M_Ratio  16
Automorph, written on right of Gram matrix:  
-9  -32
-16  -57
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ 



jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle 1 13 -13

  0  form              1          13         -13


           1           0
           0           1

To Return  
           1           0
           0           1

0  form   1 13 -13   delta  -1     ambiguous  
1  form   -13 13 1   delta  13     ambiguous  
2  form   1 13 -13


  form   1 x^2  + 13 x y  -13 y^2 

minimum was   1rep   x = 1   y = 0 disc 221 dSqrt 14  M_Ratio  196
Automorph, written on right of Gram matrix:  
-1  -13
-1  -14
=========================================

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle 1 7 -3

  0  form              1           7          -3


           1           0
           0           1

To Return  
           1           0
           0           1

0  form   1 7 -3   delta  -2     ambiguous  
1  form   -3 5 3   delta  2
2  form   3 7 -1   delta  -7
3  form   -1 7 3   delta  2     ambiguous            -1 composed with form zero  
4  form   3 5 -3   delta  -2
5  form   -3 7 1   delta  7
6  form   1 7 -3


  form   1 x^2  + 7 x y  -3 y^2 

minimum was   1rep   x = 1   y = 0 disc 61 dSqrt 7  M_Ratio  49
Automorph, written on right of Gram matrix:  
-79  -585
-195  -1444
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ 

$\endgroup$
  • $\begingroup$ This is very interesting! I suppose then that the number of small representable integers by an indefinite binary quadratic form are then intimately related to the period of the continued fraction? $\endgroup$ – Stanley Yao Xiao Jun 7 at 18:49
  • $\begingroup$ @StanleyYaoXiao seems likely, but I do not know how tight the relationship would be. There was an original definition of "reduced" for indefinite forms, perhaps that gives an upper bound on the number of reduced forms in the cycle $\langle a, ? , ?? \rangle $ for a fixed integer $a.$ SEE ALSO my mathoverflow.net/questions/264315/… $\endgroup$ – Will Jagy Jun 7 at 20:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.