On the fast Khintchine spectrum in continued fractions
| Autor: | Ai-Hua, Fan, Liao, Lingmin, Wang, Bao-Wei, Wu, Jun |
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| Rok vydání: | 2012 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $\psi$. Comment: 10 pages |
| Databáze: | arXiv |
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