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On approximation by Lüroth series. (English) Zbl 0870.11039
It is well-known that each \(x\in(0,1]\) can be expressed in the form of its Lüroth expansion \[ x= \frac{1}{a_1}+ \frac{1}{v_1a_1}+\cdots+ \frac{1}{v_na_{n+1}}+\cdots, \] where \(a_n\geq 2\) \((n=1,2,\dots)\) are integers and \(v_n=a_1(a_1-1)\dots a_n(a_n-1)\) \((n=1,2,\dots)\) [cf. J. Galambos, Representations of real numbers by infinite series, Lect. Notes Math. 502, 66-69 (1976; Zbl 0322.10002)]. Put \(p_n/q_n= 1/a_1+ \sum_{k=1}^{n-1} (1/v_ka_{k+1})\) \((n=1,2,\dots)\). H. Jager and C. de Vroedt [Nederl. Akad. Wet., Proc., Ser. A 72, 31-42 (1969; Zbl 0167.32201)] proved some fundamental results about the stochastic variables \(a_i= a_i(x)\) \((i=1,2,\dots)\).
In this paper these results are extended to the study of functions \[ \vartheta_n= \vartheta_n(x)= q_n/\bigl|x-{\textstyle{\frac{p_n}{q_n}}} \bigr|\qquad (n=1,2,\dots). \] It is proved here that if \(t\in (0,1]\) then for almost all \(x\) we have \[ \lim_{N\to\infty} \frac{1}{N}\bigl|\{j\leq N: \vartheta_j(x)<t\}\bigr|= F(t),\quad \text{where}\quad F(t)= \sum_{k=2}^{[\frac{1}{t}]+1} \frac{t}{k}+ \frac{1}{[\frac{1}{t}]+1} \] and \(|M|\) denotes the number of elements of \(M\). Several metric results are obtained concerning the functions \((\vartheta_n, \vartheta_{n+1})\), \(\vartheta_n+\vartheta_{n+1}\) and \(\vartheta_n< \vartheta_{n+1}\). The correlation coefficient for the correlation between \(\vartheta_n\) and \(\vartheta_{n+1}\) is given, as well.

MSC:
11J70 Continued fractions and generalizations
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
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