Abstract
Let \([a_1(x),a_2(x),\ldots ,a_n(x),\ldots ]\) be the continued fraction expansion of an irrational number \(x\in (0,1)\), and \(q_n(x)\) be the denominator of its n-th convergent. In this note, the Baire category of the set
for \(\alpha ,\beta \in [0,\infty ]\) with \(\alpha \le \beta \) is studied. We prove that the set \(E(\alpha ,\beta )\) is residual if and only if \(\alpha =0\) and \(\beta =\infty \).
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Acknowledgements
The research is supported by the Fundamental Research Funds for the Central Universities (No. 30922010809), GuangDong Basic and Applied Basic Research Foundation (No. 2022A1515111189) and China Postdoctoral Science Foundation (No. 2023M734060).
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Chang, X., Dong, Y., Liu, M. et al. Baire category and the relative growth rate for partial quotients in continued fractions. Arch. Math. 122, 41–46 (2024). https://doi.org/10.1007/s00013-023-01914-6
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DOI: https://doi.org/10.1007/s00013-023-01914-6