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Asymptotic behavior for the sum of partial quotients in continued fraction expansions
Part of:
Probabilistic theory: distribution modulo $1$; metric theory of algorithms
Classical measure theory
Real functions
Published online by Cambridge University Press: 11 September 2025
Abstract
Let 
$[a_1(x),a_2(x),a_3(x),\dots ]$ be the continued fraction expansion of an irrational number 
$x\in (0,1)$. Denote by 
$S_{n}(x):=\sum _{k=1}^{n} a_{k}(x)$ the sum of partial quotients of x. From the results of Khintchine (1935), Diamond and Vaaler (1986), and Philipp (1988), it follows that for Lebesgue almost every 
$x \in (0,1)$, We investigate the Baire category and Hausdorff dimension of the set of points for which the above limit inferior and limit superior assume any prescribed values. We also conduct analogous analyses for the sum of products of consecutive partial quotients.
$$\begin{align*}\liminf _{n \rightarrow \infty} \frac{S_{n}(x)}{n \log n}=\frac{1}{\log 2} \quad \text {and} \quad \limsup _{n \rightarrow \infty} \frac{S_{n}(x)}{n \log n}=\infty. \end{align*}$$
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- © The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society
Footnotes
This research was supported by the Natural Science Foundation of Jiangsu Province (Grant No. BK20241541) and the National Undergraduate Training Program for Innovation and Entrepreneurship (Grant No. 202410288085Z).
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