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On the application of dependence with complete connections to the metrical theory of G-continued fractions. Dependence with complete connections. (English) Zbl 0644.10035
Lith. Math. J. 27, No. 1, 32-40 (1987) and Lit. Mat. Sb. 27, No. 1, 68-79 (1987).
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.
Reviewer: J.Galambos

##### MSC:
 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
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