We show that the growth rate of denominator $Q_n$ of the $n$-th convergent of negative expansion of $x$ and the rate of approximation: $$ \frac{ \log{ n}}{ n}\log{ \left|x-\frac{ P_n}{ Q_n}\right|}\rightarrow -\frac{ \pi^2}{ 3} \quad \text{ in measure.} $$ for a.e. $x$. In the course of the proof, we reprove known inspiring results that arithmetic mean of digits of negative continued fraction converges to 3 in measure, although the limit inferior is 2, and the limit superior is infinite almost everywhere.
Categories: math.DS math.NT