On the metrical theory of continued fractions

Abstract:

Suppose ${b_k}$ denotes either $\phi (k)$ or $\phi ({p_k})\;(k = 1,2, \ldots )$ where the polynomial $\phi$ maps $\mathbb {N}$ to $\mathbb {N}$ and ${p_k}$ denotes the $k$th rational prime. Suppose $({c_k}(x))_{k = 1}^\infty$ denotes the sequences of partial quotients of the continued function expansion of the real number $x$. Then for certain functions $F:{\mathbb {R}_{ \geqslant 0}} \to \mathbb {R}$ we show that \[ \lim \limits _{N \to \infty } {F^{ - 1}}\left [ {\frac {{F({c_{{b_1}}}(x)) + \cdots + F({c_{{b_k}}}(x))}} {N}} \right ] = {F^{ - 1}}\left [ {\frac {1} {{(\log 2)}}\int _0^1 {\frac {{F({c_1}(x))}} {{1 + x}}dx} } \right ]\] almost everywhere with respect to Lebesgue measure. This result with ${b_k} = k$ is classical and due to Ryll-Nardzewski.