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For our traditional representation of numbers we can study properties of the digits such as normality: Normal number (Wikipedia).

This is easily extended to other bases but how about the rather different continued fractions: Continued fraction (Wikipedia)?

Is there an analogue of normality? We cannot ask that the "digits" are equally distributed as there is no limit on their values. We could however study their distribution. Has this been attempted? Is there some distribution that almost all numbers conform to?

Additional: I made some calculations on the frequencies of the digits in continued expressions for numbers in the range: $[0,1)$. However, the distribution is not consistent and depends on the preceding digits. So, I did not obtain a simple expected distribution for the digits.

Jam
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badjohn
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  • Previous related questions: https://math.stackexchange.com/questions/2109606/typicality-of-boundedness-of-entries-of-continued-fraction-representations and https://math.stackexchange.com/questions/112671/unique-continued-fraction – Gerry Myerson Apr 23 '19 at 04:25

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The Gauss-Kuzmin Theorem says that for almost all reals $x$ and for all positive integers $k$ the proportion of partial quotients of $x$ equal to $k$ is $$-\log_2\left(1-{1\over(k+1)^2}\right)$$

Gerry Myerson
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This is indeed something that people do (and have looked at). For me, the most notable example comes from J. Vandehey in Non-trivial matrix actions preserve normality for continued fractions and New normality constructions for continued fraction expansions. In this case, normality requires the blocks of partial quotients appear with frequency based on the Gauss measure of the corresponding cylinder set.

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