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Asymptotic behavior for the sum of partial quotients in continued fraction expansions
- Part of
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- Journal:
- Canadian Mathematical Bulletin , First View
- Published online by Cambridge University Press:
- 11 September 2025, pp. 1-13
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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.
$$\begin{align*}\liminf _{n \rightarrow \infty} \frac{S_{n}(x)}{n \log n}=\frac{1}{\log 2} \quad \text {and} \quad \limsup _{n \rightarrow \infty} \frac{S_{n}(x)}{n \log n}=\infty. \end{align*}$$
