A quantitative version of the theorem on Khintchine's constant

Abstract

In the paper we provide measure estimates for the set of numbers whose sequence of products of continued fraction partial quotients $M_n = a_1 \ldots a_n$ has exponential growth with rate close to the one predicted by Khintchine's theorem, i.e. for which \begin{equation*} e^{(\kappa - T)n} \leqslant M_n \leqslant e^{(\kappa + T)n} \end{equation*} for a fixed $T > 0$ and all $n$ greater than some fixed integer $N$, where $e^\kappa = 2.685\ldots$ is the Khintchine constant. Choosing $N$ large enough the measure can be made arbitrarily close to full, for any given $T$. The bounds are not of asymptotic nature, but explicit in terms of the parameters involved. In the proof we compile several known result of large deviations theory, employing the cumulant method in particular. We also discuss the numerical values of the quantities involved.