Given a strictly positive integer $A$, let $D(A)$ denote the set of all 
real quadratic algebraic numbers with a continued fraction having almost all coefficients 
$\leq A$.


Consider the field $Q\_A$ generated by all elements of $D(A)$. One has $Q\_1=\mathbb Q[\sqrt{5}]\subset Q\_2\subset Q\_3,\dots$.


The inclusion $Q\_1\subset Q\_2$ is strict since $Q\_2$ contains for example $\sqrt{2}=[1;2,2,2,\dots]$ and $\sqrt{3}=[1;1,2,1,2,\dots]$.


Are there other strict inclusions? Are there cases of equality? (I ignore for example if 
$Q\_2$ is a proper subfield of the field $\mathbb Q[\sqrt{\mathbb N}]$
generated by all real quadratic number-fields.)


A related question: Given a real quadratic algebraic number 
$\alpha$ with continued fraction expansion $\alpha=[a\_0;a\_1,a\_2,\dots]$
consider the mean value $\mu(\alpha)=\lim\_{n\rightarrow\infty}\frac{1}{n}\sum\_{j=1}^na\_j$.
Since $[a\_0;a\_1,\dots]$ is ultimately periodic, this mean value is a well-defined rational number $\geq 1$ if $\alpha$ is irrational. 


Are there examples of quadratic number-fields $\mathbb Q[\sqrt{N}]$
such that $\inf\_{\alpha\in \mathbb Q[\sqrt{N}]\setminus\mathbb Q}\mu(\alpha)>1$?



Coefficients in the periodic part of continued fractions for real quadratic algebraic numbers
In [this paper of McMullen](http://dx.doi.org/10.1112/S0010437X09004102), he asks on p. 842 whether all real quadratic fields
contain infinitely many continued fractions with coefficients bounded by 2? In this case, one
would have $Q\_d=Q\_2=\mathbb{Q}(\sqrt{\mathbb{N}})$ for all $d\geq 2$.
See also [McMullen's slides](http://www.math.harvard.edu/~ctm/expositions/home/text/papers/cf/slides/slides.pdf) from his talk at MSRI in February.


