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Limit theorems related to beta-expansion and continued fraction expansion. (English) Zbl 1408.11076

Summary: Let \(\beta > 1\) be a real number and \(x \in [0, 1)\) be an irrational number. Denote by \(k_n(x)\) the exact number of partial quotients in the continued fraction expansion of \(x\) given by the first \(n\) digits in the \(\beta\)-expansion of \(x\) (\(n \in \mathbb{N}\)). In this paper, we show a central limit theorem and a law of the iterated logarithm for the random variables sequence \(\{ k_n,\ n \geq 1 \}\), which generalize the results of C. Faivre [Arch. Math. 70, No. 6, 455–463 (1998; Zbl 0921.11041)] and J. Wu [Monatsh. Math. 153, No. 1, 83–87 (2008; Zbl 1136.11050)] respectively from \(\beta = 10\) to any \(\beta > 1\).

MSC:

11K50 Metric theory of continued fractions
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
60F05 Central limit and other weak theorems
60F15 Strong limit theorems
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References:

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