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Apologies if this has already been asked here, but I wasn’t able to find it. The Gauss-Kuzmin distribution is the asymptotic probability distribution of the coefficients of the continued fractions of almost all real numbers:

$$p(k)=-\log_{2}\left(1-\frac{1}{(1+k)^{2}}\right)$$

All proofs of this result and similar results that I have encountered use ergodic theory (e.g. here and here and here) which I am not very familiar with. I would really like to get some understanding of this formula though, since I believe that from it one can derive formulae like Khinchin's constant (e.g. according to this answer). I was wondering if anyone knew of and could provide a proof of this result that does not use ergodic theory, perhaps even just more basic real or complex analysis and number theory?

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    $\begingroup$ You can find the proof without ergodic theory in Khinchins book-Continued Fractions. Kuzmin's original proof doesn't use ergodic theory explicitly. $\endgroup$
    – D.G
    Commented Dec 1, 2019 at 14:22

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