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arXiv · 1208.1825

On the fast Khintchine spectrum in continued fractions

Fan Ai-Hua·Lingmin Liao·Bao-Wei Wang·Jun Wu
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Abstract

For $x\in [0,1)$, let $x=[a_1(x), a_2(x),...]$ 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 $\psi(n)/n\to \infty$ as $n\to \infty$. In this note, the fast Khintchine spectrum, i.e., the Hausdorff dimension of the set $$ E(\psi):=\Big{x\in [0,1): \lim_{n\to\infty}\frac{1}{\psi(n)}\sum_{j=1}^n\log a_j(x)=1\Big} $$ is completely determined without any extra condition on $\psi$.

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Fan Ai-Hua, Lingmin Liao, Bao-Wei Wang, Jun Wu. 2012-08-09. On the fast Khintchine spectrum in continued fractions. https://arxiv.org/abs/1208.1825

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