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

TL;DR

This paper investigates the multifractal properties of digit growth rates in Schneider's p-adic continued fractions, determining Hausdorff dimensions of sets characterized by digit growth behaviors.

Contribution

It provides a complete Hausdorff dimension characterization of digit growth rate sets in p-adic continued fractions for arbitrary growth functions.

Findings

Hausdorff dimension of sets with specified digit growth rates is explicitly determined.

Dimensions of intersection sets with prescribed liminf and limsup of digit ratios are calculated.

Results extend multifractal analysis to p-adic continued fraction systems.

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 $0\leq\alpha_1<\alpha_2\leq\infty$.