Your Favourite Application of the Birkhoff Ergodic Theorem

Solution 1:

For almost every $x \in [0,1]$, the elements of the continued fraction expansion of $x$ are unbounded.

This can be seen by studying $\mathbb{R}/\mathbb{Z}$ with $\frac{1}{\log 2}\frac{1}{1+x}dx$ as the measure. Let $T: \mathbb{R}/\mathbb{Z} \to \mathbb{R}/\mathbb{Z}$ be $Tx = \{\frac{1}{x}\}$, the fractional part of $\frac{1}{x}$ (with $T0 := 0$), and $f(x) = \lfloor \frac{1}{x}\rfloor$. The ergodic theorem says that, if $x = [x_1,x_2,\dots]$, then $$\frac{x_1+x_2+\dots+x_N}{N} = \frac{1}{N}\sum_{n=0}^{N-1} f(T^n x) \to \int_0^1 f(y)\frac{1}{\log 2}\frac{dy}{1+y} = +\infty.$$ This tells us a bit more than unboundedness, but when I first heard that the result is proven by ergodic theory, I was shocked how this analytic machinery could prove this very number theoretic statement [I didn't know much analytic number theory at the time either :P]. Of course "almost every" appears in the statement of the result, but I definitely don't view that as a big restriction. This result is what made me want to study ergodic theory.