Acta mathematica scientia,Series A ›› 2024, Vol. 44 ›› Issue (4): 847-858.

### On Hausdorff Dimension of the Exceptional Sets of Partial Maximal Digits for Lüroth Expansion

Chen Junyou(),Zhang Zhenliang*()

1. School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331
• Received:2023-07-18 Revised:2024-02-25 Online:2024-08-26 Published:2024-07-26
• Supported by:
Science and Technology Research Program of Chongqing Municipal Education Commission(KJQN202100528);Natural Science Foundation of Chongqing(CSTB2022NSCQMSX-1255)

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.

CLC Number:

• O156.7
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