# Distribution of the computable numbers on the real number line

If we order all the positive computable real numbers $r_1,r_2,r_3...$ by their Kolmogorov complexity in some language $L$, then make a histogram plot of the $r_i$ on the real line, and we scale it such that the height at one point is constant, and let the bin-size go to 0, what can be said about the resulting distribution as $n\rightarrow\infty$? I'd expect it to approach something (very roughly) like the normal distribution.

Is there any nontrivial $L$ for which we know of a distribution that becomes apparent for large $n$?

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Certainly not like the normal distribution. For instance, consider the Ackermann function. $A(n,n)$ takes constant+log n bits to express in any reasonable language, so the probability of a number at least as big as $A(n,n)$ will be much higher than $e^{- A(n,n)^2}$, which is a reasonable estimate for the normal distribution. Therefore, the distribution has longer tails than the normal distribution. My guess is that for most reasonable languages, a sufficient portion of the probability mass will go off to infinity, and that most facts about the distribution will not be computable. – Will Sawin Nov 24 '11 at 12:24

For any Turing-complete language, the class of descriptions of the form "$r$ is an integer with these digits:" followed by the digits, will have a constant positive probability. But this is exactly a uniform distribution on the first $2^n$ integers, which "goes to infinity". Hence, the limit object will not be a distribution (will not have total mass 1).