# A quantitative version of the theorem on Khintchine's constant

## TL;DR

This paper provides explicit measure estimates for the set of numbers with continued fraction partial quotients' products growing exponentially at a rate close to Khintchine's constant, using large deviations theory and cumulant methods.

## Contribution

It offers a quantitative, non-asymptotic measure estimate for numbers with continued fraction products near Khintchine's predicted growth rate, with explicit bounds.

## Key findings

- Measure estimates can be made arbitrarily close to full for large N.
- Bounds are explicit and not asymptotic, depending on parameters.
- Employs large deviations theory and cumulant method in proof.

## 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.