Abstract
For \(x\in [0,1)\), let \(x=[a_1(x), a_2(x),\ldots ]\) be its continued fraction expansion with partial quotients \(\{a_n(x), n\ge 1\}\). Let \(\psi : \mathbb{N } \rightarrow \mathbb{N }\) be a function with \(\psi (n)/n\rightarrow \infty \) as \(n\rightarrow \infty \). In this note, the fast Khintchine spectrum, i.e., the Hausdorff dimension of the set
is completely determined without any extra condition on \(\psi \). This fills a gap of the former work in Fan et al. (Ergod Theor Dyn Syst 29:73–109, 2009).
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The authors express their sincere thanks to the referees for their detailed suggestions and comments. This work is partially supported by PICS program No. 5727 and NSFC Nos. 11171124 and 11225101.
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Communicated by H. Bruin.
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Fan, A., Liao, L., Wang, B. et al. On the fast Khintchine spectrum in continued fractions. Monatsh Math 171, 329–340 (2013). https://doi.org/10.1007/s00605-013-0530-1
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DOI: https://doi.org/10.1007/s00605-013-0530-1