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What is a good reference for the proof of Khinchin-Levy theorem on continued fractions, using the full power of ergodic theory? A google search is not quite yielding the needed stuff.

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You can find the ergodic-theoretic proof in Einsiedler & Ward's "Ergodic Theory with a view towards Number Theory". Chapter 3 discusses continued fractions and investigates asymptotic properties of them using methods of ergodic theory. Corollary 3.8 contains the theorem you mention.

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