Czechoslovak Mathematical Journal, first online, pp. 1-24
Hausdorff dimension of some exceptional sets in Lüroth expansions
Ao Wang, Xinyun Zhang
Received March 19, 2024. Published online March 19, 2025.
Abstract: We study the metrical theory of the growth rate of digits in Lüroth expansions. More precisely, for $ x\in( 0,1 ]$, let $[ d_1( x ) ,d_2 ( x) ,\cdots]$ denote the Lüroth expansion of $x$. We completely determine the Hausdorff dimension of the sets $E_{\sup} ( \psi) = \biggl\{ x\in( 0,1 ] \colon\limsup_{n\rightarrow\infty} \frac{\log d_n ( x)}{\psi( n )}=1 \biggr\}$, $E ( \psi) = \biggl\{ x\in( 0,1 ] \colon\lim_{n\rightarrow\infty} \frac{\log d_n ( x)}{\psi( n )}=1 \biggr\}$ and $E_{\inf} (\psi) =\biggl\{ x\in( 0,1 ] \colon\liminf_{n\rightarrow\infty} \frac{\log d_n ( x )}{\psi( n)}=1 \biggr\}$, where $\psi\colon\mathbb{N} \rightarrow\mathbb{R} ^+ $ is an arbitrary function satisfying $ \psi( n ) \rightarrow\infty$ as $n\rightarrow\infty$.
Affiliations: Ao Wang, School of Mathematics and Statistics, Huazhong University of Science and Technology, 1037, Luoyu Road, HongShan District, Wuhan, Hubei 430074, P. R. China, e-mail: wang_ao17@163.com; Xinyun Zhang (corresponding author), School of Mathematics and Information Science, Nanchang Hangkong University, 696, Fenghe South Avenue, Honggutan New District, Nanchang, Jiangxi 330063, P. R. China, e-mail: 71371@nchu.edu.cn
