For x∈[0,1)\documentclass[12pt]{minimal}
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\begin{document}$${x \in [0,1)}$$\end{document}, let Sn(x) denote the summation of the first n digits in the dyadic expansion of x and rn(x) denote the run-length function. Let H denote the set of monotonically increasing functions φ:N→(0,+∞)\documentclass[12pt]{minimal}
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\begin{document}$${\varphi \colon \mathbb{N} \to (0, +\infty)}$$\end{document} with limn→∞φ(n)=+∞\documentclass[12pt]{minimal}
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\begin{document}$${\lim_{n\to\infty}
\varphi(n)=+\infty}$$\end{document}. For any φ∈H\documentclass[12pt]{minimal}
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\begin{document}$${\varphi \in H}$$\end{document} and 0≤α≤1\documentclass[12pt]{minimal}
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\begin{document}$${0 \leq \alpha \leq 1}$$\end{document}, we prove that the set
x∈[0,1]:limn→∞Sn(x)n=α,lim infn→∞rn(x)φ(n)=0,lim supn→∞rn(x)φ(n)=+∞\documentclass[12pt]{minimal}
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\begin{document}$$\left\{x\in [0,1]: \lim_{n\to \infty}
\frac{S_n(x)}{n}=\alpha, \liminf_{n\to\infty}
\frac{r_n(x)}{\varphi(n)}=0, \limsup _{n\to\infty}
\frac{r_n(x)}{\varphi(n)}=+\infty\right\}$$\end{document}either has Hausdorff dimension H(α)/log2\documentclass[12pt]{minimal}
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\begin{document}$${H(\alpha)/{\rm log} 2}$$\end{document} or is empty. Here H(α)=-αlogα-(1-α)log(1-α)\documentclass[12pt]{minimal}
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\begin{document}$${H(\alpha) = -{\alpha}{\rm log} \alpha-(1-\alpha){\rm log}(1-\alpha)}$$\end{document} is the classical entropy function.
