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Keywords:
Lüroth expansion; exceptional set; Borel-Bernstein theorem; Hausdorff dimension
Lüroth expansion; exceptional set; Borel-Bernstein theorem; Hausdorff dimension
Summary:
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 $$ \begin{aligned} E_{\sup } ( \psi ) = & \biggl \{ x\in ( 0,1 ] \colon \limsup _{n\rightarrow \infty } \frac {\log d_n ( x)}{\psi ( n )}=1 \biggr \} ,\\ E ( \psi ) = & \biggl \{ x\in ( 0,1 ] \colon \lim _{n\rightarrow \infty } \frac {\log d_n ( x)}{\psi ( n )}=1 \biggr \} \end{aligned} $$ 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 $.
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