On the application of dependence with complete connections to the metrical theory of G-continued fractions. Dependence with complete connections (Q1102321)

Every irrational x in the interval [G-2, G], with \(G=(1+\sqrt{5})/2\), has a continued fraction expansion of the form \(x=\epsilon_ 1/(\alpha_ 1+\epsilon_ 2/(\alpha_ 2+...\), where \(\epsilon_ j\) is either -1 or \(+1\), and each digit \(\alpha_ j\) is an odd positive integer. Via the theory of random systems with complete connections the author establishes a number of deep results on the digits \(\alpha_ j\), \(j\geq 1\), and on the approximants \(p_ n/q_ n=\epsilon_ 1/(\alpha_ 1+\epsilon_ 2/(\alpha_ 2+...+\epsilon_ n/\alpha_ n)...)\) including the limit relation, as \(n\to +\infty\), \(\lim n^{-1} \log | x-p_ n/q_ n| =-\pi^ 2/9 \log G\) almost everywhere.