Multifractal analysis of the growth rate of digits in Schneider's $p$-adic continued fraction dynamical system

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 $\{a_n(x){\}}_{n\ge 1}$ from the viewpoint of multifractal analysis. The Hausdorff dimension of the set $$E_{\sup }(\psi )=\{x\in p{\mathbb{Z}}_p:\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{a_n(x)}{\psi (n)}=1\}$$ is completely determined for any $\psi :\mathbb{N}\to {\mathbb{R}}^{+}$ satisfying $\psi (n)\to \mathrm{\infty }$ as $n\to \mathrm{\infty }$. As an application, we also calculate the Hausdorff dimension of the intersection sets $$E_{\inf }^{}(\psi ,{\alpha }_1,{\alpha }_2)=\{x\in p{\mathbb{Z}}_p:\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}{\displaystyle \frac{a_n(x)}{\psi (n)}}={\alpha }_1,\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}{\displaystyle \frac{a_n(x)}{\psi (n)}}={\alpha }_2\}$$ for the above function $\psi$ and $0\le {\alpha }_1<{\alpha }_2\le \mathrm{\infty }$.