Farei deries and continued fractions to the nearest even number

V. I. Parusnikov


Abstract: The evenness of the numerator and denominator of convergents of continued fraction to the nearest even number is opposite. We establish there that for nearly by $1/3$ of almost all real numbers, convergents of its regular continued fraction expansion have either even numerator and odd denominator and eiter even denominator and odd numirator. This property holds for both the lebesque measure just as for the distribution function of the so called Minkowski question mark function $?(x)$. Incindentally we establish that for the measure that coresponds to the distribution $?(x)$, the mean of the $n$ first elements of regular continued fraction tends to 2 as $n$ tends to infinity.