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Is one of those claims about continued fractions true ?

$1)$ Almost all real numbers have a continued fraction representation with a bounded sequence of entries.

$2)$ Almost all real numbers have a continued fraction representation with an unbounded sequence of entries.

Intuitively I would expect that the unbounded sequence is the "typical" outcome. How can I determine the measure of both types of continued-fraction-sequences ?

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Almost all real number have continued fractions expansion which is not only unbounded, but distribute according to the Gauss-Kuzmin measure. This follows from the fact that the Gauss map is ergodic. See for example here. This result should also appear in every textbook about ergodic theory and\or continued fractions (for example "ergodic theory with a view towards Number Theory" by Einsiedler and Ward has a whole chapter on this).

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