Khinchin's Constant - Sketch of Proof

Sketch of Proof

The proof presented here was arranged by Ryll-Nardzewski (1951) and is much simpler than Khinchin's original proof which did not use ergodic theory.

Since the first coefficient a₀ of the continued fraction of x plays no role in Khinchin's theorem and since the rational numbers have Lebesgue measure zero, we are reduced to the study of irrational numbers in the unit interval, i.e., those in . These numbers are in bijection with infinite continued fractions of the form, which we simply write, where a₁, a₂, ... are positive integers. Define a transformation T:I → I by

The transformation T is called the Gauss–Kuzmin–Wirsing operator. For every Borel subset E of I, we also define the Gauss–Kuzmin measure of E

Then μ is a probability measure on the σ-algebra of Borel subsets of I. The measure μ is equivalent to the Lebesgue measure on I, but it has the additional property that the transformation T preserves the measure μ. Moreover, it can be proved that T is an ergodic transformation of the measurable space I endowed with the probability measure μ (this is the hard part of the proof). The ergodic theorem then says that for any μ-integrable function f on I, the average value of is the same for almost all :

Applying this to the function defined by f = log(a₁), we obtain that

for almost all in I as n → ∞.

Taking the exponential on both sides, we obtain to the left the geometric mean of the first n coefficients of the continued fraction, and to the right Khinchin's constant.