Hausdorff dimension of some exceptional sets in Lüroth expansions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929327941091328 |
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| author | Wang, Ao Zhang, Xinyun |
| author_facet | Wang, Ao Zhang, Xinyun |
| contents | In this paper, we study the metrical theory of the growth rate of digits in Lüroth expansions. More precisely, for $ x\in \left( 0,1 \right] $, let $ \left[ d_1\left( x \right) ,d_2\left( x \right) ,\cdots \right] $ denote the Lüroth expansion of $ x $, we completely determine the Hausdorff dimension of the following sets \begin{align*}
E_{\mathrm{sup}}\left( ψ\right) =\Big\{ x\in \left( 0,1 \right] :\limsup\limits_{n\rightarrow \infty}\frac{\log d_n\left( x \right)}{ψ\left( n \right)}=1 \Big\} ,
\end{align*} \begin{align*}
E\left( ψ\right) =\Big\{ x\in \left( 0,1 \right] :\lim_{n\rightarrow \infty}\frac{\log d_n\left( x \right)}{ψ\left( n \right)}=1 \Big\}
\end{align*} and \begin{align*}
E_{\mathrm{inf}}\left( ψ\right) =\Big\{ x\in \left( 0,1 \right] : \liminf_{n\rightarrow \infty}\frac{\log d_n\left( x \right)}{ψ\left( n \right)}=1 \Big\} , \end{align*} where $ ψ:\mathbb{N} \rightarrow \mathbb{R} ^+ $ is an arbitrary function satisfying $ ψ\left( n \right) \rightarrow \infty$ as $n\rightarrow \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_17135 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hausdorff dimension of some exceptional sets in Lüroth expansions Wang, Ao Zhang, Xinyun Number Theory In this paper, we study the metrical theory of the growth rate of digits in Lüroth expansions. More precisely, for $ x\in \left( 0,1 \right] $, let $ \left[ d_1\left( x \right) ,d_2\left( x \right) ,\cdots \right] $ denote the Lüroth expansion of $ x $, we completely determine the Hausdorff dimension of the following sets \begin{align*} E_{\mathrm{sup}}\left( ψ\right) =\Big\{ x\in \left( 0,1 \right] :\limsup\limits_{n\rightarrow \infty}\frac{\log d_n\left( x \right)}{ψ\left( n \right)}=1 \Big\} , \end{align*} \begin{align*} E\left( ψ\right) =\Big\{ x\in \left( 0,1 \right] :\lim_{n\rightarrow \infty}\frac{\log d_n\left( x \right)}{ψ\left( n \right)}=1 \Big\} \end{align*} and \begin{align*} E_{\mathrm{inf}}\left( ψ\right) =\Big\{ x\in \left( 0,1 \right] : \liminf_{n\rightarrow \infty}\frac{\log d_n\left( x \right)}{ψ\left( n \right)}=1 \Big\} , \end{align*} where $ ψ:\mathbb{N} \rightarrow \mathbb{R} ^+ $ is an arbitrary function satisfying $ ψ\left( n \right) \rightarrow \infty$ as $n\rightarrow \infty$. |
| title | Hausdorff dimension of some exceptional sets in Lüroth expansions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2404.17135 |