The growth speed for the product of consecutive digits in Lüroth expansions

Abstract

For $x\in [0,1),$ let $[d_{1}(x),d_{2}(x),\ldots ]$ be its Lüroth expansion and $\big \{\frac{p_{n}(x)}{q_{n}(x)}, n\ge 1\big \}$ be the sequence of convergents of x. For $\alpha , \beta \in [0,+\infty )$ with $\alpha \le \beta ,$ we define the exceptional sets

$$\begin{aligned} E(\beta )=\Big \{x\in [0,1):\limsup _{n\rightarrow \infty }\frac{\log \big ( d_{n}(x)d_{n+1}(x)\big )}{\log q_{n}(x)}=\beta \Big \} \end{aligned}$$

and

$$\begin{aligned} F(\alpha ,\beta )= & {} \Big \{x\in [0,1):\liminf _{n\rightarrow \infty }\frac{\log \big ( d_{n}(x)d_{n+1}(x)\big )}{\log q_{n}(x)}=\alpha ,\\&\quad \limsup _{n\rightarrow \infty }\frac{\log \big ( d_{n}(x)d_{n+1}(x)\big )}{\log q_{n}(x)}=\beta \Big \}. \end{aligned}$$

In this paper, we completely determine the Hausdorff dimension of sets $E(\beta )$ and $F(\alpha ,\beta ).$