Open problems in continued fractions theory I propose to collect here open problems from the theory of continued fractions. Any types of continued fractions are welcome.
 A: From M. Waldschmidt, "Open Diophantine Problems" (Moscow Mathematical Journal vol. 4, no. 1, 2004, pp. 245-305):


*

*Does there exist a real algebraic number of degree $\geq 3$ with bounded partial quotients?

*Does there exist a real algebraic number of degree $\geq 3$ with unbounded partial quotients?

A: Guy, Unsolved Problems In Number Theory, F21, attributes to Bohuslav Divis the conjecture that in each real quadratic field there is an irrational with all partial quotients 1 or 2; more generally, same question but with 1 and 2 replaced by any pair of distinct positive integers. 
A: Find fixed points (mod 1) of Minkowski's question mark function, see A058914 from "The On-Line Encyclopedia of Integer Sequences". This picture  
is taken from http://mathworld.wolfram.com/MinkowskisQuestionMarkFunction.html 
It shows that there is only one positive fixed point (mod 1) less than 1/2. It is approximately 0.42037. Does this constant have a closed-form expression? 
A: 
Hermite's problem is an open problem in mathematics posed by Charles Hermite in 1848. He asked for a way of expressing real numbers as sequences of natural numbers, such that the sequence is eventually periodic precisely when the original number is a cubic irrational.

from https://en.wikipedia.org/wiki/Hermite%27s_problem
A: Guy, Unsolved Problems In Number Theory, F21, attributes to Leo Moser the conjecture that there is a constant $c$ such that every $n$ can be expressed as $n=a+b$ in such a way that the sum of the partial quotients of $a/b$ is less than $c\log n$. 
A: It is known that (see J.W. Porter, On a theorem of
Heilbronn)
$$H(b)=\dfrac{1}{\varphi(b)}\sum\limits_{1\le a\le b\atop(a,b)=1}\ell(a/b)=
\dfrac{2\log 2}{\zeta(2)}\cdot\log
b+C_P-1+O_\varepsilon(b^{-1/6+\varepsilon}),$$ where
$\ell(a/b)$ is a length of standart continued fraction
expansion and $C_P$ is Porter's constant. Averaging
over numerators and denominators one can prove  asymptotic
formula for the variance. Let
$$E(R)=\dfrac{2}{R(R+1)}\sum\limits_{b\le
R}\sum\limits_{a\le b}\ell(a/b)
$$
and $${D}(R)=\dfrac{2}{R(R+1)}\sum\limits_{b\le
R}\sum\limits_{a\le b}\left( \ell(a/b)-E(R)\right)^2.
$$ Then (see D. Hensley, The Number of Steps in the Euclidean Algorithm)
$${D}(R)=D_1\cdot\log R+o(\log R).
$$
But if denominator is fixed then for the variance only
right oder bound is known (see Bykovskii V.A., Estimate
for dispersion of lengths of continued fractions):
$$\dfrac{1}{b}\sum\limits_{a=1}^{b}\left(\ell\left(\dfrac{a}{b}\right)-
\dfrac{2\log2}{\zeta(2)}\log b\right)^2\ll\log b.$$

Conjecture:
  $$\dfrac{1}{\varphi(b)}\sum\limits_{1\le a\le b\atop(a,b)=1}(\ell(a/b)-H(b))^2=D_1\log b+o(\log b).$$

A: A collection of Open problems in geometry of continued fractions by Oleg Karpenkov.
A: Zaremba's Conjecture:

every integer appears as the denominator of a finite continued fraction whose partial quotients are bounded by an absolute constant.

