Diophantine approximation by negative continued fraction

Abstract

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: $$\frac{\log n}{n}\log \left|x-\frac{P_n}{Q_n}\right|\to -\frac{{\pi }^2}{3}\text{in measure.}$$ 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.

8 pages