Title: Multifractal analysis of the growth rate of digits in Schneider's $p$-adic continued fraction dynamical system Show Chinese title

Title: 分形分析施耐德的$p$-进制连分数动力系统的数字增长速率

Abstract: 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 \[E_{\sup}(\psi)=\Big\{x\in p\mathbb{Z}_p:\ \limsup\limits_{n\to\infty}\frac{a_n(x)}{\psi(n)}=1\Big\}\] is completely determined for any $\psi:\mathbb{N}\to\mathbb{R}^{+}$ satisfying $\psi(n)\to \infty$ as $n\to\infty$. As an application, we also calculate the Hausdorff dimension of the intersection sets \[E^{\sup}_{\inf}(\psi,\alpha_1,\alpha_2)=\left\{x\in p\mathbb{Z}_p:\liminf_{n\rightarrow\infty}\dfrac{a_n(x)}{\psi(n)}=\alpha_1,~\limsup_{n\rightarrow\infty}\dfrac{a_n(x)}{\psi(n)}=\alpha_2\right\}\] for the above function $\psi$ and $0\leq\alpha_1<\alpha_2\leq\infty$. Show Chinese abstract

Abstract: 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. 该集合\[E_{\sup}(\psi)=\Big\{x\in p\mathbb{Z}_p:\ \limsup\limits_{n\to\infty}\frac{a_n(x)}{\psi(n)}=1\Big\}\]的豪斯多夫维数对于任何满足$\psi(n)\to \infty$的$\psi:\mathbb{N}\to\mathbb{R}^{+}$都完全确定，其值为$n\to\infty$。 作为应用，我们还计算了上述函数$\psi$和$0\leq\alpha_1<\alpha_2\leq\infty$的交集集合\[E^{\sup}_{\inf}(\psi,\alpha_1,\alpha_2)=\left\{x\in p\mathbb{Z}_p:\liminf_{n\rightarrow\infty}\dfrac{a_n(x)}{\psi(n)}=\alpha_1,~\limsup_{n\rightarrow\infty}\dfrac{a_n(x)}{\psi(n)}=\alpha_2\right\}\]的豪斯多夫维数。