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Recall Khinchin's result, recently and less recently brought to center stage.

To each real number $x$ is associated a simple continued fraction $[a_o; a_1, a_2, \dots ]$, where $a_0\in\Bbb Z$ and $a_i, i\geq 1$ is a positive integer. Then for almost all $x$ the sequence $\left(\prod_{i=1}^{n} a_i\right)^{1/n}$ converges to a constant $K$ that doesn't depend on $x$.

If you look carefully, real numbers are required to state this result only because they bring the notion of "almost all" that makes it work. Apart from that, it could be just a result about sequences of integers (especially given that the continued fraction $[a_o; a_1, a_2, \dots ]$ does converge to a real number for any sequence of positive integers).

So my question is:

What are other natural measures on the set of sequences of positive integers? What about the limits of $\left(\prod_{i=1}^{n} a_i\right)^{1/n}$ with respect to these measures?

Arnaud Mortier
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  • Of course you get a good limit when the $a_n$ are IID. But the remarkable thing is that, for the continued fraction, even though the $a_n$ are not independent, we get a limit. For other non-independent cases, see ergodic theory https://en.wikipedia.org/wiki/Ergodic_theory – GEdgar Dec 02 '19 at 12:46
  • @GEdgar Trying to process your comment, you're saying that a specific measure put on the set of positive integers sequences will result in a specific way to see the $a_i$ as random variables (I can roughly see how this would work), and that this is the key feature to look at? – Arnaud Mortier Dec 03 '19 at 12:37
  • "A probability measure on the space of sequences of positive integers" is the same thing as "a random sequence of positive integers", according to the standard (Kolmogorov) model used for probability theory. If you are interested in this, find a textbook on probability theory (advanced enough that it presupposed measure theory). – GEdgar Dec 03 '19 at 12:41
  • @GEdgar There are no probability measures in the OP, the original example is the Lebesgue measure. But of course finite volume measures are not excluded. – Arnaud Mortier Dec 03 '19 at 12:44
  • Arnaud: And the $x$ used in the original example could be taken as $x \in (0,1)$, where Lebesgue measure is a probability measure. – GEdgar Dec 03 '19 at 12:48

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