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Khinchin showed that given the simple continued fraction of a real number,

$$r = a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1} {\ddots}}}$$

then it is almost always true that the partial quotients $a_i$ satisfy,

$$K = \lim_{n \rightarrow \infty } \left( a_1 a_2 ... a_n \right) ^{1/n} =2.685452\dots$$

where $K$ is Khinchin's constant. (Some exceptional $r$ are the rationals, roots of quadratic equations, and rational powers of $e$.)

Q1: Given $n$, let $T_n$ be the total number of partial quotients $a_i = 1$. Is it almost always true that, $$H=\lim_{n \rightarrow \infty } \frac{T_n}{n} = 0.415\dots$$ exists and converges? If it does, can $H$ be expressed in terms of $K$ and other constants?

Numerical evidence for various transcendental and algebraic constants are given below with $n=10^k$ and entries as $T_n$:

$$\begin{array}{|c|c|c|c|c|} \hline \text{constant}&10^3&10^4&10^5&10^6&10^7\\ \hline \pi&412& 4206& 41494& 414526& 4148280\\ \Gamma\big(\tfrac{1}{2}\big)&417& 4178& 41620& 415352& 4151849\\ \log(2)&433& 4148& 41430& 415443&-\\ \log(3)&429& 4170& 41458& 414919&-\\ T&396& 4084& 41172& 414458&-\\ P&410& 4087& 41364& 415180&-\\ K&418& 4111& 41379&-&-\\ C&412& 4147& 41543&-&-\\ \zeta(3)&418& 4223&-&-&-\\ \hline \end{array}$$

where $T, P, K, C$ are the tribonacci, plastic, Khinchin, and Catalan constants.

Q2: Anybody able to fill in the blanks? Or extend it to $n>10^{7}$ so we can have more decimal digits of $H$? (I know the continued fraction for $\pi$ has been computed to more than $n>10^{10}$ terms.)

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    $\begingroup$ @TheUserWhoWasConcerned about the sample space: It usually works quite well, as these "traditional" real numbers (algebraic, transcendental,...) are defined in very different ways, usually uncorrelated with the property you are testing, so you actually sample quite a wide range of number classes. However, testing with numbers, specifically defined through their continued fraction expansion, would be problematic :) Of course, numerical testing is an inspiration, sanity check and quick and dirty review of what properties do we even have. At the end, you must still do some real math :) $\endgroup$
    – orion
    Jul 14, 2015 at 10:19

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A probability distribution of the continued fraction expansion terms follows the Gauss-Kuzmin distribution for almost all irrational numbers:

$$p(k)=-\log_2\left(1-\frac{1}{(k+1)^2}\right)=\log_2\frac{(k+1)^2}{k(k+2)}$$

All generalized Khinchin's constants (including $K=K_0$, the geometric mean), are derived from this distribution.

In this case, you seek the fraction of terms with value $1$, which is

$$p(1)=\log_2 \frac{4}{3}\approx 0.4150375$$

So it turns out this constant you observed is expressible with elementary functions.

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  • $\begingroup$ Beautiful! So satisfying to see the trend I observed is in fact converging to a definite value, and $H = p(1) \approx 0.415037\dots$ has a closed-form. :) $\endgroup$
    – Tito Piezas III
    Jul 14, 2015 at 15:34

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