On the dimension of certain sets araising in the base two expansion
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866908803274899456 |
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| author | Neunhäuserer, Jörg |
| author_facet | Neunhäuserer, Jörg |
| contents | We show that for the base two expansion \[ x=\sum_{i=1}^{\infty}2^{-(d_{1}(x)+d_{2}(x)+\dots+d_{i}(x))}\] with $x\in(0,1]$ and $d_{i}(x)\in\mathbb{N}$ the set $A=\{x|\lim_{i\to\infty}d_{i}(x)=\infty\}$ has Hausdorff dimension zero, this is opposed to a result on the continued fraction expansion, here $A$ has Hausdorff dimension $1/2$, see \cite{[GO]}. Furthermore we construct subsets of $B=\{x|\limsup_{i\to\infty}d_{i}(x)=\infty\}$ which have Hausdorff dimension one and find a dimension spectrum in set $B$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2201_09641 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the dimension of certain sets araising in the base two expansion Neunhäuserer, Jörg Dynamical Systems 11K55, 28A80 We show that for the base two expansion \[ x=\sum_{i=1}^{\infty}2^{-(d_{1}(x)+d_{2}(x)+\dots+d_{i}(x))}\] with $x\in(0,1]$ and $d_{i}(x)\in\mathbb{N}$ the set $A=\{x|\lim_{i\to\infty}d_{i}(x)=\infty\}$ has Hausdorff dimension zero, this is opposed to a result on the continued fraction expansion, here $A$ has Hausdorff dimension $1/2$, see \cite{[GO]}. Furthermore we construct subsets of $B=\{x|\limsup_{i\to\infty}d_{i}(x)=\infty\}$ which have Hausdorff dimension one and find a dimension spectrum in set $B$. |
| title | On the dimension of certain sets araising in the base two expansion |
| topic | Dynamical Systems 11K55, 28A80 |
| url | https://arxiv.org/abs/2201.09641 |