Recall that for a real number $\alpha$, it has a continued fraction expansion usually written as

$$\displaystyle \alpha = [a_0; a_1, a_2, \cdots].$$

Moreover, $\alpha$ is rational if and only if its continued fraction expansion is finite, and is a quadratic irrational if and only if the continued fraction expansion is eventually periodic. The numbers $a_0; a_1, a_2, \cdots$ are called *partial quotients* of $\alpha$.

It is known that with respect to Lebesgue measure, almost all real numbers $\alpha$ have the property that the partial quotients of $\alpha$ follow a specific distribution: the frequency that $k$ appears in the sequence $\{a_0; a_1, a_2, \cdots\}$ is $\frac{1}{\log 2} \log \left(\frac{(k+1)^2}{k(k+2)} \right)$.

Consider the interval $I_X = (X, 2X]$ for some (large) positive number $X$. For each integer $n \in I_X$, consider the continued fraction expansion of $\sqrt{n}$, say $\sqrt{n} = [a_0; a_1, a_2, \cdots]$. We know that this sequence is eventually periodic, and in fact $\sqrt{n} = [a_0; \overline{a_1, a_2, \cdots, a_2, a_1, 2a_0}]$ where the bar denotes the periodic part and the string $a_1, a_2, \cdots, a_2, a_1$ denotes a palindrome.

My question is: as $n$ runs over $I_X$, how are the partial quotients of $\sqrt{n}$ distributed? In other words, for each positive integer $k$ what is the frequency of appearance of $k$ as partial quotients of $\sqrt{n}$ as $n$ runs over $I_X$?

Note that the frequency is 0 if $k \gg X^{1/2}$, since it is known that the partial quotients of $\sqrt{n}$ are at most $2 \sqrt{n}$ in size.