Abstract:

For any $x\in (0,1)$, let $x=[ d_{1}(x), d_{2} (x), \cdots, d_{n} (x)]$ be its Lüroth expansion. Denote the maximal digits of the first $n$ digits by $L_{n}(x)=\max \left\{d_{1}(x), \cdots, d_{n}(x)\right\}.$ For any real number $0< \alpha < \beta < \infty$, we determine the Hausdorff dimension of the exceptional set

$F_{\phi}(\alpha, \beta)=\left\{x \in(0,1): \liminf _{n \rightarrow \infty} \frac{L_{n}(x)}{\phi(n)}=\alpha, \limsup _{n \rightarrow \infty} \frac{L_{n}(x)}{\phi(n)}=\beta\right\},$

where $\phi (n)= n^{\gamma} (\gamma>0)$ or ${\rm e}^{n^{\gamma}} (\gamma>0 )$. This supplements the results of [13]. Similarly, the corresponding exceptional sets of the sums of digits in Lüroth expansion are also studied.

Key words: Lüroth expansion, Largest digit, Hausdorff dimension