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 π = {x0, x1, x2, . . . xn}, where x's are the continued fraction elements computed with an exact value of π up to N precision. We numerically compute probability distribution for the elements and observe a striking power-law behavior P (x) ∼ 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.