The average denominator of the continued fraction expansion of $\pi$.

Question

I was interested in the long term behavior of continued fraction denominators, so I plotted the average of the first $n$ terms in the continued fraction expansion of $\pi$ as a function of $n$ and got the following graph:

And it turns out that at the $453294$th position, we suddenly get a $12996958$ among many one and two digit numbers. Why the sudden large number?

I know there are other continued fractions with strange behavior, notably the Champernowne's Numbers. $C_{10}$ seems especially weird, as its continued fraction expansion begins with:

$$ [0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, 4 57540 11139 10310 76483 64662 82429 56118 59960 39397 10457 55500 06620\\ 04393 09026 26592 56314 93795 32077 47128 65631 38641 20937 55035 52094 60718 30899\\ 84575 80146 98631 48833 59214 17830 10987, 6, 1, 1, 21, 1, 9, 1, 1, 2, 3, 1, 7, 2, 1]$$

What is known about the asymptotic behavior of continued fractions? Why do such large terms appear between smaller ones? Is there a measure of how "erratic" the expansion is, such as the limit of the variance of the denominators?

EDIT: Fixed reason for jump.

Answer 1

There is a theorem from real analysis using the ideas of ergodic theory that states, for almost every real number $x \in \mathbb R$, the natural number $n$ appears in the continued fraction expansion of $x$ with frequency $\log_{2} \left ( \frac{(n+1)^{2}}{n(n+2)} \right )$.

To prove this fact, consider the Gauss measure $\nu (E) = \frac{1}{\log(2)} \int_{0}^{1} \frac{1}{1+t} dt$, on $([0,1]\setminus \mathbb Q)$. Also consider the transformation $U(x)= \{\frac{1}{x}\}$, where $\{\}$ denotes fractional part. This transformation is ergodic, and from here proving the theorem is a straightforward application of the ergodic theorem.

Answer 2

For almost every real number, the geometric mean of the denominators is equal to Khinchin's constant, about $2.6854$, if we can believe that Wikipedia page.

The article mentions that $e$ is an exception, and that's easy if one knows its simple continued fraction expansion.

The article says:

Among the numbers whose geometric mean of the coefficients $a_i$ in the continued fraction expansion apparently (based on numerical evidence) tends to Khinchin's constant are $\pi$, the Euler–Mascheroni constant $\gamma$, and Khinchin's constant itself. However, none of these limits has been rigorously established,

One could wonder whether for almost every real number, the asymptotic distribution of the denominators is the same.

Answer 3

Not sure about pi, but about Champernowne's number (CN from now on), it's not terribly hard to see why it has such huge numbers near the beginning of its continued fraction.

For example, CN is 0.12345679 when rounded to eight decimals, just like 10/81, which equals 10 (1/9)^2 = 1/10 + 2/100 + 3/1000 + 4/10,000 + ...
This explains the first few coefficients. However, CN isn't periodic. After the first nine decimals, CN consists of the numbers 10 to 99, then 100 to 999 and so on.

Now, the number 0.01020304050607080910111213...9799000102(repeating a 198-digit sequence)... equals 100/9891, the square of 10/99, which has a much simpler decimal expansion, = 0.101010...
One can see that CN is quite well approximated by a rather simple combination of the two quantities 10/81 and 100/9801, and that approximation is good for almost 189 digits, where CN has exhausted all possible 2-digit sequences.
[0;8,9,1,149083,1,1,1,4,1,1,1,3,4,1,1,1,15] is probably a number 0.123456789 followed by a repeating part 198 digits in length and starting with 10111213...9091929394959697...
And even if it's not, the length is probably 180 to 200 digits, with the same initial digits given above.

So, after a few small coefficients, one moderately big one, and another few small ones, we encounter a coefficient with close to 180 digits.
Likewise, one would expect a coeffient with almost 2700 digits that approximates CN up to the end of the 900 different 3-digit sequences, another with close to 36000 digits that includes most of the 4-digit sequences, and so on. CN can therefore generate huge coefficients rather soon in its continued fraction expansion.

That's not a rigorous proof, but the chance that a significantly smaller coefficient (let's say, at most half the predicted digits) would do at some point is negligibly small, even with only a coarse statistical argument.

Most of my reasoning seems to hold for any base > 2, so I'd challenge the notion that C10 is "especially weird", at least among other Champernowne numbers. But that could be an ambiguous wording in the question. (If you use a different base than 10, there should be one where my argument can be replicated for any given Champernowne number except maybe C2.)