Scaling in simple continued fraction

Abstract

We consider a class of real numbers, a subset of irrational numbers and certain mathematical constants, for which the elements in the simple continued fraction appears to be random. As an illustrative example, one can consider $\pi = \{x_0, x_1, x_2, \dots x_n\}$, where $x$'s are the continued fraction elements computed with an exact value of $\pi$ up to $N$ precision. We numerically compute probability distribution for the elements and observe a striking power-law behavior $P(x)\sim x^{-2}$. The statistical analysis indicates that the elements are uncorrelated and the scaling is robust with respect to the precision. Our arguments reveal that the underlying mechanism generating such a scaling may be sample space reducing process.