Two Constants: Khinchin and Chaitin

Take a real number, $x$.  Write out its continued fraction:

$\displaystyle{x=a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\cdots}}}}$

It’s an intriguing fact that if you look at the sequence of geometric means $a_0, (a_0a_1)^{1/2}, (a_0a_1a_2)^{1/3}, \ldots$ this approaches a single constant, called Khinchin’s constant, which is approximately $K\approx 2.69$, for almost every $x$.  This means that if you were to pick $x$ (for convenience, say it’s between 0 and 1) by writing a decimal point and then repeatedly rolling a ten-sided die forever to generate the digits after that, the $x$ you generate would have this property with probability 1.

However, as the Wikipedia page above says, although almost all $x$ have this property (call it the “Khinchin property”), no number that wasn’t specifically constructed to have the Khinchin property has been proven to do so (and some numbers, like $e$ and $\sqrt{2}$ and all rational numbers, have been shown to not have the Khinchin property).

If you want to be the first to find a number $x$ having the Khinchin property that wasn’t specifically constructed to have it, my advice is to try Chaitin’s Constant, $\Omega$.  Roughly, you can think of $\Omega$ as the probability that a randomly selected Turing machine will halt, although there are a few more technicalities than that.

More importantly for our purposes, it’s very likely to have the Khinchin property, because it’s algorithmically random, meaning it has all computable properties almost all numbers have! That means that the following statement implies that $\Omega$ has the Khinchin property:

There is a computable function $\delta\colon \mathbb{N}\times\mathbb{N}\to\mathbb{N}$ satisfying:  For all $n,m\in\mathbb{N}$, the set of numbers $x$ between 0 and 1 such that $|(a_0\cdots a_{N-1})^{1/N} - K| < 2^{-m}$ for all $N > \delta(n,m)$ has measure at least $1-2^{-n}$.

Proving that sounds a lot easier to me that, e.g., proving $\pi$ has the Khinchin property (note that the fact that $\delta$ is computable is the hard part).

However, some might quibble about whether or not this meets the original criterion: it’s definitely true that $\Omega$ wasn’t constructed to have the Khinchin property; however, in a certain sense, it was constructed to have every such property!