Let [a1(x),a2(x),a3(x),…]\documentclass[12pt]{minimal}
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\begin{document}$$[a_1(x),a_2(x),a_3(x),\ldots ]$$\end{document} be the continued fraction expansion of an irrational number x∈(0,1)\documentclass[12pt]{minimal}
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\begin{document}$$x\in (0,1)$$\end{document}. It is known that for Lebesgue almost all x∈(0,1)\Q\documentclass[12pt]{minimal}
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\begin{document}$$x\in (0,1)\setminus {\mathbb {Q}}$$\end{document}, lim infn→∞logan(x)logn=0andlim supn→∞logan(x)logn=1.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \liminf _{n \rightarrow \infty } \frac{\log a_n(x)}{\log n}=0 \ \ \ \text {and}\ \ \ \limsup _{n \rightarrow \infty } \frac{\log a_n(x)}{\log n}=1. \end{aligned}$$\end{document}In this note, the Baire classification and Hausdorff dimension of E(α,β):=x∈(0,1)\Q:lim infn→∞logan(x)logn=α,lim supn→∞logan(x)logn=β\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} E(\alpha ,\beta ):=\left\{ x\in (0,1)\setminus {\mathbb {Q}}: \liminf _{n \rightarrow \infty } \frac{\log a_n(x)}{\log n}=\alpha ,\ \limsup _{n \rightarrow \infty } \frac{\log a_n(x)}{\log n}=\beta \right\} \end{aligned}$$\end{document}for all α,β∈[0,∞]\documentclass[12pt]{minimal}
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\begin{document}$$\alpha , \beta \in [0,\infty ]$$\end{document} with α≤β\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \le \beta $$\end{document} are studied. We prove that E(α,β)\documentclass[12pt]{minimal}
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\begin{document}$$E(\alpha ,\beta )$$\end{document} is residual if and only if α=0\documentclass[12pt]{minimal}
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\begin{document}$$\alpha =0$$\end{document} and β=∞\documentclass[12pt]{minimal}
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\begin{document}$$\beta =\infty $$\end{document}, and the Hausdorff dimension of E(α,β)\documentclass[12pt]{minimal}
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\begin{document}$$E(\alpha ,\beta )$$\end{document} is as follows: dimHE(α,β)=1,α=0;1/2,α>0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \dim _{\textrm{H}} E(\alpha ,\beta ) = \left\{ \begin{array}{ll} 1, &{} {\alpha =0;} \\ 1/2, &{} {\alpha >0.} \end{array} \right. \end{aligned}$$\end{document}Moreover, the Hausdorff dimension of the intersection of E(α,β)\documentclass[12pt]{minimal}
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\begin{document}$$E(\alpha ,\beta )$$\end{document} and the set of points with non-decreasing partial quotients is also provided.
