Some exceptional sets of Borel–Bernstein theorem in continued fractions

Abstract

Let \([a_1(x),a_2(x), a_3(x),\ldots ]\) denote the continued fraction expansion of a real number \(x \in [0,1)\). This paper is concerned with certain exceptional sets of the Borel–Bernstein Theorem on the growth rate of \(\{a_n(x)\}_{n\geqslant 1}\). As a main result, the Hausdorff dimension of the set

$$\begin{aligned} E_{\sup }(\psi )=\left\{ x\in [0,1):\ \limsup \limits _{n\rightarrow \infty }\frac{\log a_n(x)}{\psi (n)}=1\right\} \end{aligned}$$

is determined, where \(\psi :{\mathbb {N}}\rightarrow {\mathbb {R}}^+\) tends to infinity as \(n\rightarrow \infty \).

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Acknowledgements

We would like to thank Professor Lingmin Liao for his valuable comments. We also sincerely thank the referees for their helpful suggestions and remarks.

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Correspondence to Kunkun Song.

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This research was supported by National Natural Science Foundation of China (11771153, 11801591, 11971195) and Science and Technology Program of Guangzhou (202002030369). Kunkun Song would like to thank China Scholarship Council (CSC) for financial support (201806270091)

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Fang, L., Ma, J. & Song, K. Some exceptional sets of Borel–Bernstein theorem in continued fractions. Ramanujan J (2020). https://doi.org/10.1007/s11139-020-00320-8

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Keywords

  • Continued fractions
  • Partial quotients
  • Hausdorff dimension

Mathematics Subject Classification

  • 11K50
  • 28A80