Multifractal analysis of the growth rate of digits in Schneider's $p$-adic continued fraction dynamical system
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909223251607552 |
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| author | Song, Kunkun Wu, Wanlou Yu, Yueli Zeng, Sainan |
| author_facet | Song, Kunkun Wu, Wanlou Yu, Yueli Zeng, Sainan |
| contents | Let $\mathbb{Z}_p$ be the ring of $p$-adic integers and $a_n(x)$ be the $n$-th digit of Schneider's $p$-adic continued fraction of $x\in p\mathbb{Z}_p$. We study the growth rate of the digits $\{a_n(x)\}_{n\geq1}$ from the viewpoint of multifractal analysis. The Hausdorff dimension of the set \[E_{\sup}(ψ)=\Big\{x\in p\mathbb{Z}_p:\ \limsup\limits_{n\to\infty}\frac{a_n(x)}{ψ(n)}=1\Big\}\] is completely determined for any $ψ:\mathbb{N}\to\mathbb{R}^{+}$ satisfying $ψ(n)\to \infty$ as $n\to\infty$. As an application, we also calculate the Hausdorff dimension of the intersection sets \[E^{\sup}_{\inf}(ψ,α_1,α_2)=\left\{x\in p\mathbb{Z}_p:\liminf_{n\rightarrow\infty}\dfrac{a_n(x)}{ψ(n)}=α_1,~\limsup_{n\rightarrow\infty}\dfrac{a_n(x)}{ψ(n)}=α_2\right\}\] for the above function $ψ$ and $0\leqα_1<α_2\leq\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09081 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multifractal analysis of the growth rate of digits in Schneider's $p$-adic continued fraction dynamical system Song, Kunkun Wu, Wanlou Yu, Yueli Zeng, Sainan Number Theory Dynamical Systems Let $\mathbb{Z}_p$ be the ring of $p$-adic integers and $a_n(x)$ be the $n$-th digit of Schneider's $p$-adic continued fraction of $x\in p\mathbb{Z}_p$. We study the growth rate of the digits $\{a_n(x)\}_{n\geq1}$ from the viewpoint of multifractal analysis. The Hausdorff dimension of the set \[E_{\sup}(ψ)=\Big\{x\in p\mathbb{Z}_p:\ \limsup\limits_{n\to\infty}\frac{a_n(x)}{ψ(n)}=1\Big\}\] is completely determined for any $ψ:\mathbb{N}\to\mathbb{R}^{+}$ satisfying $ψ(n)\to \infty$ as $n\to\infty$. As an application, we also calculate the Hausdorff dimension of the intersection sets \[E^{\sup}_{\inf}(ψ,α_1,α_2)=\left\{x\in p\mathbb{Z}_p:\liminf_{n\rightarrow\infty}\dfrac{a_n(x)}{ψ(n)}=α_1,~\limsup_{n\rightarrow\infty}\dfrac{a_n(x)}{ψ(n)}=α_2\right\}\] for the above function $ψ$ and $0\leqα_1<α_2\leq\infty$. |
| title | Multifractal analysis of the growth rate of digits in Schneider's $p$-adic continued fraction dynamical system |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2406.09081 |