Diophantine Approximation by Negative Continued Fraction

Abstract

We are interested in the statistical behavior of a certain continued fraction whose associated dynamical system has the infinite invariant measure. We show the growth rate of denominator $Q_n$ of the $n$-th convergent of negative continued fraction 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 $x$. In the course of the proof, we reprove related 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.