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If $x$ is a uniformly random number in $[0,1]$, what distribution should the $n$-th term in its continued fraction expansion follow?

What is the expected vale of $a_n$ in $[a_0;a_1,a_2,\dots]$?

Here is the expansion for $\pi$.

What does it say about a number if there is some regularity in the sequence? Why does $e$ have regularity, but $\pi$ apparently does not?

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up vote 3 down vote accepted

The Gauss-Kuzmin distribution is what you are looking for.

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Of course $a_0 = 0$ if $x \in (0,1)$. For $n \ge 1$, $\frac{1}{3k^2} \le\Pr(a_n = k) \le \frac{2}{k^2}$ (see e.g. Khinchin, "Continued Fractions", sec. 12). In particular, $E[a_n] = \infty$.

I assume you know that the expansion terminates iff $x$ is rational and is eventually periodic iff $x$ is a quadratic irrational. If you know something about how the $a_n$ grow, there are consequences about how well $x$ can be approximated by rationals. Otherwise, I don't know what can be said in general about $x$ from the fact that there is "some regularity".

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