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On the central limit theorem for random variables related to the continued fraction expansion. (English) Zbl 0857.11038

Applications of ergodic theory in the theory of continued fractions are based on the transformation \(T:[0,1] \to[0,1]\), \(T(0)=0\), \(T(x)= {1\over x} - [{1 \over x}]\) for \(x=[0;a_1(x), a_2(x), \dots] \in(0,1]\). It is well-known that \(([0,1],T,\nu)\) is an ergodic system, where \(\nu\) denotes the Gauss measure on \([0,1]\) defined by the density \(h(x)=(1/ \log 2) (1/(1+x))\).
Unfortunately there are many important sequences of random variables that are not expressible in the form \(f(T^{n-1}(x))\) \((n=1,2, \dots)\) for a suitable \(f\). Such a sequence is \((\Theta_n (x))^\infty_{n=1}\), where \(|x-p_n/q_n |= \Theta_n(x)/q^2_n\) and \(p_n/q_n\) \((n=1,2, \dots)\) are convergents of \(x\). Define the mapping \[ W:[0,1]' \times [0,1] \to[0,1]' \times[0,1]; (x,y)\mapsto \left(T(x), {1\over a_1(x)+y} \right), \] where \([0,1]'\) denotes the set of all irrational numbers of \([0,1]\). Then \(\Theta_n(x)\) and also \(r_n(x)\) can be expressed in the form \(f(W^n(x),0)\) by a suitable function \(f\); here, \[ r_n(x) = \left|\left. x-{p_n \over q_n} \right|\right/ \left|x-{p_{n-1} \over q_{n-1}} \right|\qquad (n=1,2,\dots). \] The mapping \(W\) preserves the probability measure on \([0,1]' \times[0,1]\) defined by \[ d \mu (x,y) = {1\over\log 2} {dx dy \over(1+xy)^2} \] and \((W,\mu)\) is an ergodic system [cf. H. Nakada, Tokyo J. Math. 4, 399-426 (1981; Zbl 0479.10029)]. In the paper [W. Bosma, H. Jager and F. Wiedijk, Indag. Math. 45, 281-299 (1983; Zbl 0519.10043)] it is shown that \(\theta_n\) and \(r_n\) satisfy the strong law of large numbers.
In this paper the central limit theorem for the random variables \(X_n(x) = f(W^n(x,t))\) is proved, where \(t\) is a fixed number in \([0,1]\).

MSC:

11K50 Metric theory of continued fractions
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