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Some metrical observations on the approximation by continued fractions. (English) Zbl 0519.10043

In this paper some metric results on regular continued fractions are proved. The starting point for obtaining these results is Lenstra’s conjecture. Let \(p_n/q_n\), \((n = 1,2,\ldots)\) be the continued fraction convergents of the irrational number \(x\). It is well-known that \(\vert x - p_n/q_n\vert < 1/q_n^2\). Hence we have \(\vert x - p_n/q_n\vert = \theta_n(x)/q_n\), \(0< \theta_n(x) <1\), \(\theta_n(x) = q_n \vert q_nx - p_n\). According to Lenstra’s conjecture for each \(z\in [0,1] \) for almost all \(x\) we have \[ \liminf_{n\to\infty} \frac1n \#\{j\le n; \theta_j(x)\le z\} = F(z), \] where \[ F(z) = \begin{cases} \frac{z}{\log 2} &\qquad\text{for } 0\le z\le\frac12, \\ \frac{1}{\log 2} (-z + \log 2z + 1) &\qquad \text{for } \frac12\le z\le 1. \tag{1} \end{cases} \] D. E. Knuth obtained the following result: If \(E_n(z) = \{x; \theta_n(x)\le z\}\), then \(m(E_n(z)) = F(z)+ O(g^n)\), \(g = \tfrac12 (\sqrt 5 - 1)\) \((m(H)\) denotes the Lebesgue measure of the set \(H)\). The authors outline the proof of Lenstra’s conjecture based on the previous result of D. E. Knuth. In this proof the ergodicity of the mapping \(Tx = 1/x - [1/x]\) \((x\in (0,1))\) with respect to the Gauss measure is used.
Further they give another proof of Lenstra’s conjecture. Let \(f: \mathbb N\to [0,1]\) be an arithmetical function. Denote by \(A(n,z)\) the number of integers \(j\) with \(1\le j\le n\), \(f(j)\le z\). Put \(\nu_n(z) = A(n,z)/n\) \((n=1,2,\ldots)\). Then \(\nu_n\) is a distribution function on \([0,1]\). If there exists a distribution function \(G\) on \([0,1]\) such that \(\displaystyle \lim_{n\to\infty} \nu_n(z) = G(z)\) for each continuity point \(z\) of \(G\), then \(f\) is said to have the limiting distribution \(G\). Denote by \(\theta(x)\) the arithmetical function: \(n\mapsto \theta_n(x)\). Then for almost all \(x\) the function \(\theta(x)\) has the limiting distribution \(F\) given in (1).
For the proof of this fact the authors use some properties of the operator \(T^*: M\to M\), \(M = ((0,1) - \mathbb Q) \times (0,1)\), \(\mathbb Q\) is the set of all rational numbers, \(T^*(x,y) = (Tx,1/([1/x] +y))\). The proof is based on the fact that \((M,\mathcal B,\mu,T^*)\) is an ergodic system [cf. I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic theory. New York etc.: Springer-Verlag (1982; Zbl 0493.28007), p. 241; H. Nakada, Tokyo J. Math. 4, 399–426 (1981; Zbl 0479.10029)]. Here \(\mathcal B\) denotes the class of all Borel subsets of \(M\) and \[ \mu(E) = \frac1{\log 2} \iint_E \frac{dx\,dy}{(1 +xy)^2} \quad\text{for } E\in\mathcal B. \] In the paper limiting distributions of some further arithmetical functions connected with regular continued fractions are determined. E.g. the limiting distribution of the function \(Q(x): n\to q_{n-1}/q_n\) is \(F(z) = \log(1 + z)/\log 2\) \((0\le z\le 1)\).
Similarly the limiting distribution of the following functions are given: \[ r(x): n \mapsto \vert x - p_n/q_n\vert / \vert x - p_{n-1}/q_{n-1}; \] \[ d(x): n \mapsto d_n(x), \vert x - p_n/q_n\vert = d_(x)/q_nq_{n+1}, \text{ etc}. \] At the end the authors extend these considerations for Nakada’s expansions (loc. cit.) based upon the operator \[ f_\alpha: [\alpha - 1, \alpha] \rightarrow [\alpha - 1, \alpha] , \quad f_\alpha(x) = \vert 1/x\vert - [\vert 1/x\vert + 1 - \alpha], \] \(x\ne 0\), \(\tfrac12\le \alpha\le 1\) (for \(\alpha = 1\) we obtain the usual regular continued fractions).

MSC:

11K65 Arithmetic functions in probabilistic number theory
11K50 Metric theory of continued fractions
11J70 Continued fractions and generalizations