Limit theorems for sub-sums of partial quotients of continued fractions. (English) Zbl 1423.11136
Summary: This paper studies the limit behaviour of sums of the form
\[ T_n(x) = \sum_{1 \leq j \leq n} c_{k_j}(x),\qquad (n = 1, 2,\ldots) \]
where \((c_j(x))_{j \geq 1}\) is the sequence of partial quotients in the regular continued fraction expansion of the real number \(x\) and \((k_j)_{j \geq 1}\) is a strictly increasing sequence of natural numbers. Of particular interest is the case where for irrational \(\alpha\), the sequence \((k_j \alpha)_{j \geq 1}\) is uniformly distributed modulo one and \((k_j)_{j \geq 1}\) is good universal. It was observed by the second author, for this class of sequences \((k_j)_{j \geq 1}\) that we have \(\lim_{n \rightarrow \infty} \frac{T_n(x)}{n} = + \infty\) almost everywhere with respect to Lebesgue measure. The case \(k_j = j(j = 1, 2, \ldots)\) is classical and due to A. Ya. Khinchin. Building on work of H. Diamond, Khinchin, W. Philipp, L. Heinrich, J. Vaaler and others, in the special case where \(k_j = j(j = 1, 2, \ldots,)\) we examine the asymptotic behaviour of the sequence \((T_n(x))_{n \geq 1}\) in more detail.
\[ T_n(x) = \sum_{1 \leq j \leq n} c_{k_j}(x),\qquad (n = 1, 2,\ldots) \]
where \((c_j(x))_{j \geq 1}\) is the sequence of partial quotients in the regular continued fraction expansion of the real number \(x\) and \((k_j)_{j \geq 1}\) is a strictly increasing sequence of natural numbers. Of particular interest is the case where for irrational \(\alpha\), the sequence \((k_j \alpha)_{j \geq 1}\) is uniformly distributed modulo one and \((k_j)_{j \geq 1}\) is good universal. It was observed by the second author, for this class of sequences \((k_j)_{j \geq 1}\) that we have \(\lim_{n \rightarrow \infty} \frac{T_n(x)}{n} = + \infty\) almost everywhere with respect to Lebesgue measure. The case \(k_j = j(j = 1, 2, \ldots)\) is classical and due to A. Ya. Khinchin. Building on work of H. Diamond, Khinchin, W. Philipp, L. Heinrich, J. Vaaler and others, in the special case where \(k_j = j(j = 1, 2, \ldots,)\) we examine the asymptotic behaviour of the sequence \((T_n(x))_{n \geq 1}\) in more detail.
MSC:
| 11K55 | Metric theory of other algorithms and expansions; measure and Hausdorff dimension |
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