Hausdorff dimension of some exceptional sets in L\"{u}roth expansions

TL;DR

This paper determines the Hausdorff dimension of certain exceptional sets related to the growth rate of digits in L"uroth expansions, extending the metrical theory of these expansions for arbitrary growth functions.

Contribution

It provides a complete Hausdorff dimension characterization of sets defined by digit growth rates in L"uroth expansions for general functions diverging to infinity.

Findings

Explicit Hausdorff dimension formulas for sets with specified digit growth behaviors.

Extension of metrical theory to arbitrary diverging functions in L"uroth expansions.

Identification of the size of exceptional digit growth sets in terms of Hausdorff dimension.

Abstract

In this paper, we study the metrical theory of the growth rate of digits in L\"{u}roth 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 L\"{u}roth expansion of $ x $, we completely determine the Hausdorff dimension of the following sets \begin{align*} E_{\mathrm{sup}}\left( \psi \right) =\Big\{ x\in \left( 0,1 \right] :\limsup\limits_{n\rightarrow \infty}\frac{\log d_n\left( x \right)}{\psi \left( n \right)}=1 \Big\} , \end{align*} \begin{align*} E\left( \psi \right) =\Big\{ x\in \left( 0,1 \right] :\lim_{n\rightarrow \infty}\frac{\log d_n\left( x \right)}{\psi \left( n \right)}=1 \Big\} \end{align*} and \begin{align*} E_{\mathrm{inf}}\left( \psi \right) =\Big\{ x\in \left( 0,1 \right] : \liminf_{n\rightarrow \infty}\frac{\log d_n\left( x \right)}{\psi \left( n \right)}=1 \Big\} , \end{align*} where $ \psi :\mathbb{N} \rightarrow \mathbb{R} ^+ $ is an arbitrary function satisfying $ \psi \left( n \right) \rightarrow \infty$ as $n\rightarrow \infty$.