Take a Gauss map $G: [0,1] \longrightarrow [0,1]$ which is
$$G(x) = \frac{1}{x} \mod 1, 0<x<1$$ and $0$ if $x=0$. Let $\mu$ be the Gauss measure. For $x \in [0,1]$ let $[a_{1}(x), a_{2}(x),...]$ denote the continued fraction expansion of $x$. I would like to show that for any $\gamma>0$ we have for $\mu$-almost all $x \in [0,1]$ $$\lim_{n \longrightarrow \infty} \frac{a_{n}(x)}{n^{1+\gamma}} = 0.$$
Obviously, an application of Birkhoff's ergodic theorem is needed. I think we need to use the fact that for an ergodic measure preserving transformation $T$, with a measure $\mu$ and a function $f \in L^{1}$ we have $ \lim _{n \longrightarrow \infty} \frac{f(T^{n}(x)}{n} = 0$ for almost all $x \in \mu$. (This can be proved by considering partial sums of the form $\sum_{k=0}^{n-1} f(T^{k}(x)$).
But I don't know how to feel in the missing details - what function $f$ to choose, that this holds for all $\gamma >0$, etc. It looks like since we know $a_{n+1} = [\frac{1}{G^{n}(x)}]$ (where the square brackets denotes the integer part), we have to choose $f = \frac{[\frac{1}{x}]}{x^{\gamma}}$.
Is this right? I would appreciate any help and advice on how to make this rigorous.