On the law of the iterated logarithm for continued fractions with sequentially restricted partial quotients
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2017
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| _version_ | 1866929289809625088 |
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| author | Stadlbauer, Manuel Zhang, Xuan |
| author_facet | Stadlbauer, Manuel Zhang, Xuan |
| contents | We establish a law of the iterated logarithm (LIL) for the set of real numbers whose $n$-th partial quotient is bigger than $α_n$, where $(α_n)$ is a sequence such that $\sum 1/α_n$ is finite. This set is shown to have Hausdorff dimension $1/2$ in many cases and the measure in LIL is absolutely continuous to the Hausdorff measure. The result is obtained as an application of a strong invariance principle for unbounded observables on the limit set of a sequential iterated function system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1707_09673 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | On the law of the iterated logarithm for continued fractions with sequentially restricted partial quotients Stadlbauer, Manuel Zhang, Xuan Dynamical Systems Probability 11K50, 60F17, 37F35 We establish a law of the iterated logarithm (LIL) for the set of real numbers whose $n$-th partial quotient is bigger than $α_n$, where $(α_n)$ is a sequence such that $\sum 1/α_n$ is finite. This set is shown to have Hausdorff dimension $1/2$ in many cases and the measure in LIL is absolutely continuous to the Hausdorff measure. The result is obtained as an application of a strong invariance principle for unbounded observables on the limit set of a sequential iterated function system. |
| title | On the law of the iterated logarithm for continued fractions with sequentially restricted partial quotients |
| topic | Dynamical Systems Probability 11K50, 60F17, 37F35 |
| url | https://arxiv.org/abs/1707.09673 |