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

Chen Junyou,, Zhang Zhenliang,*

School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331

 基金资助: 重庆市教育委员会科学技术研究项目(KJQN202100528)重庆市自然科学基金(CSTB2022NSCQMSX-1255)

 Fund supported: Science and Technology Research Program of Chongqing Municipal Education Commission(KJQN202100528)Natural Science Foundation of Chongqing(CSTB2022NSCQMSX-1255)

\begin{align*}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\},\end{align*}

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.

Keywords： Lüroth expansion; Largest digit; Hausdorff dimension

Chen Junyou, Zhang Zhenliang. On Hausdorff Dimension of the Exceptional Sets of Partial Maximal Digits for Lüroth Expansion[J]. Acta Mathematica Scientia, 2024, 44(4): 847-858

## 1 引言

\begin{align*} x=x_{1}, d_{n} := d_{n} (x)=\left[ \frac{1}{x_{n}} \right]+1, x_{n+1} =\left( x_{n} - \frac{1}{d_{n}} \right) d_{n} ( d_{n} -1),\end{align*}

$x=\frac{1}{d_{1}(x)}+\frac{1}{d_{1}(x)\left(d_{1}(x)-1\right) d_{2}(x)}+\cdots+\frac{1}{d_{1}(x)\cdots d_{n-1}(x)\left(d_{n-1}(x)-1\right) d_{n}(x)}+\cdots,$

$L_{n}(x)=\max \left\{d_{1}(x), \cdots, d_{n}(x)\right\}.$

$\lim_{n\to\infty}\lambda\left\{x\in(0,1)\colon \frac{L_n(x)}{n}<y\right\}={\rm e}^{-1/y},$

$\limsup_{n\to\infty}\frac{\log L_n(x)-\log n}{\log\log n}=1,\,\,\text{且}\,\,\liminf_{n\to\infty}\frac{\log L_n(x)-\log n}{\log\log n}=0.$

Shen[12]等人研究了上述极限的例外集, 证明了对任意的$\alpha\geq0$,集合

$\left\{x\in(0,1)\colon \lim_{n\to\infty}\frac{\log L_n(x)}{\log n}=\alpha\right\}$

$\left\{x\in(0,1)\colon\limsup_{n\to\infty}\frac{\log L_n(x)}{\log n}=\alpha,\,\, \liminf_{n\to\infty}\frac{\log L_n(x)}{\log n}=\beta \right\}$

$\liminf _{n \rightarrow \infty} \frac{L_{n}(x) \log \log n}{n}=1.$

Song[13]等人讨论了上述极限的例外集的分形维数.

$\dim_{\mathrm{H}}\left\{x \in(0,1): \lim _{n \rightarrow \infty} \frac{L_{n}(x)}{n^{\gamma}}=\alpha\right\}=1.$

$\dim_{\mathrm{H}}$来表示某个集合的Hausdorff维数.

$\operatorname{dim}_{\mathrm{H}}\left\{x \in(0,1): \lim _{n \rightarrow \infty} \frac{L_{n}(x)}{a^{n^{\gamma}}}=\alpha\right\}= \begin{cases} 1,& \gamma\in(0,\frac{1}{2}),\\[3mm] \dfrac{1}{2},& \gamma\in[\frac{1}{2},+\infty). \end{cases}$

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

(1) 当$\phi (n)= n^{\gamma}\, (\gamma>0)$, 有 $\operatorname{dim}_{\mathrm{H}}F_{\phi}(\alpha, \beta)=1$;

(2) 当$\phi (n)={\rm e}^{n^{\gamma}}\, (\gamma>0 )$, 有

$\dim_{\mathrm{H}}F_{\phi}(\alpha,\beta)=\begin{cases}1,\,\,\, \gamma\in(0,\frac{1}{2}), \\[3mm] \dfrac{1}{2},\,\,\,\gamma\in[\frac{1}{2},+\infty).\end{cases}$

$I_n(d_{1},d_2,\cdots,d_n)=\left\{x\in(0,1):d_{1}(x)=d_{1},d_2(x)=d_2,\cdots,d_{n}(x)=d_{n}\right\}$

$n$ 阶基本区间,用 $I_{n}(x)$ 表示包含$x$$n 阶基本区间,用 |I| 表示区间 I 的长度. 下面引理给出了一个 n 阶基本区间的长度. 引理 2.1[3] 对于任意的 d_{1}, \cdots, d_{n} \in \mathbb{N}$$d_j\geq2,\,\,(1\leq j\leq n)$, $n$ 阶基本区间 $I_n$ 的左右端点分别为

$\frac{1}{d_{1}}+\frac{1}{d_{1}\left(d_{1}-1\right) d_{2}}+\cdots+\prod_{k=1}^{n-1} \frac{1}{d_{k}\left(d_{k}-1\right)} \frac{1}{d_{n}}$

$\frac{1}{d_{1}}+\frac{1}{d_{1}\left(d_{1}-1\right) d_{2}}+\cdots+\prod_{k=1}^{n-1}\frac{1}{d_{k}\left(d_{k}-1\right)} \frac{1}{d_{n}}+\prod_{k=1}^{n} \frac{1}{d_{k}\left(d_{k}-1\right)}.$

$\prod_{k=1}^{n} d_{k}^{-2}\leq\left|I_n\left(d_{1}, \cdots, d_{n}\right)\right|=\prod_{k=1}^{n} \frac{1}{d_{k}\left(d_{k}-1\right)} \leq \prod_{k=1}^{n}\left(d_{k}-1\right)^{-2}.$

$E_{M}:=\left\{x \in(0,1): 2 \leq d_{j}(x) \leq M,\,\, \text{对任意的} \,\,j\geq 1 \right\},$

$B(m, n)=\left\{\left(i_{1}, \cdots, i_{n}\right) \in\{2, \cdots, m\}^{n}: \sum\limits_{k=1}^{n} i_{k}=m\right\}.$

$\sum\limits_{\left(i_{1}, \cdots, i_{k}\right) \in B(m, n)} \prod_{k=1}^{n}\left(i_{k}-1\right)^{-2 s} \leq\left(\frac{25}{2}(\zeta(2 s)+2)\right)^{n} m^{-2 s},$

### 3.1 下界估计

$\lim _{n \rightarrow \infty} \frac{\log \left(c_{2}(n)-c_{1}(n)\right)}{n^{\gamma}}=0$

$\liminf _{n \rightarrow \infty} \frac{\log c_{1}(n)}{\log n}>-\infty, \limsup _{n \rightarrow \infty} \frac{\log c_{2}(n)}{\log n}<+\infty.$

$D_n=\{(\sigma_1,\sigma_2,\cdots,\sigma_n)\colon c_{1}(n)<\sigma_{j} {\rm e}^{-j^{\gamma}}<c_{2}(n),\sigma_j\geq 2,\,\,\sigma_j\in\mathbb{N},\,\,\,1\leq j\leq n\},$

\bigcup\limits_{(d_{1}, \cdots, d_{n})\in D_n}I_{n}\left(d_{1}, \cdots, d_{n}\right)$$B\left(\gamma, c_{1}(n), c_{2}(n), 1\right) 的一个覆盖. 由 (3.3) 式, 这些基本区间的个数近似为 {\rm e}^{(1-\varepsilon) \sum\limits_{1}^{n} j^{\gamma}}\leq \prod_{j=1}^{n}\left(c_{2}(n)-c_{1}(n)\right) {\rm e}^{j^{\gamma}} \leq {\rm e}^{(1+\varepsilon) \sum\limits_{1}^{n} j^{\gamma}}. 由条件 (3.2) 可知, 存在常数 m,M 使得这些基本区间的长度满足 {\rm e}^{-2\sum\limits_{1}^{n} j^{\gamma}+2mn\log n} \leq\left|I_{n}\left(d_{1}, \cdots, d_{n}\right)\right| \leq {\rm e}^{-2\sum\limits_{1}^{n} j^{\gamma}+2Mn\log n}. 因此, 通过使用区间 \left\{I_{n}\left(d_{1}, \cdots, d_{n}\right)\right\} 作为一个覆盖, 我们得到 \dim_{\mathrm{H}}B\left(\gamma, c_{1}(n), c_{2}(n), 1\right) \leq \frac{1}{2}. 下面我们来估计下界, 考虑一个均匀分布在 B\left(\gamma, c_{1}(n), c_{2}(n), 1\right) 上面的概率测度 \mu, 即 \mu (I_{n}\left(d_{1}, \cdots, d_{n}\right))=\frac{1}{\sharp D_n}. 这里符号 \sharp 表示集合所含元素的个数. 由 (3.4) 式可知 {\rm e}^{(-1-\varepsilon) \sum\limits_{1}^{n} j^{\gamma}}\leq \mu (I_{n}\left(d_{1}, \cdots, d_{n}\right) )\leq {\rm e}^{(-1+\varepsilon) \sum\limits_{1}^{n} j^{\gamma}}. 对任意的 (d_1,d_2,\cdots,d_{n-1})\in D_{n-1}, 包含在其中的所有 I_{n}\left(d_{1}, \cdots, d_{n}\right) 构成一个区间： J_{n-1}(d_1,d_2,\cdots,d_{n-1})=\bigcup_{(d_{1}, \cdots, d_{n})\in D_n}I_{n}\left(d_{1}, \cdots, d_{n}\right), 其长度为 {\rm e}^{(1-\varepsilon)n^{\gamma}}\cdot {\rm e}^{-2\sum\limits_{1}^{n} j^{\gamma}+2mn\log n} \leq\left|J_{n-1}\left(d_{1}, \cdots, d_{n-1}\right)\right| \leq {\rm e}^{(1+\varepsilon)n^{\gamma}}\cdot {\rm e}^{-2\sum\limits_{1}^{n} j^{\gamma}+2Mn\log n}. 由于在上式的指数中, 主项为 \sum\limits_{1}^{n} j^{\gamma}, 因此, 对于任意的 r \in\left(\exp \left(-2 \sum\limits_{1}^{n} j^{\gamma}\right), \exp \left(-2 \sum\limits_{1}^{n-1} j^{\gamma}\right)\right) 和任意的 x \in B\left(\gamma, c_{1}(n), c_{2}(n), 1\right), 我们可以估计 B(x, r) 的测度 \mu(B(x, r)) \approx \left\{\begin{array}{ll} r \cdot {\rm e}^{\sum\limits_{1}^{n} j^{\gamma}}, & r<{\rm e}^{-2 \sum\limits_{1}^{n} j^{\gamma}+n^{\gamma}}, \\[2mm] {\rm e}^{-\sum\limits_{1}^{n-1} j^{\gamma}}, & r>{\rm e}^{-2 \sum\limits_{1}^{n} j^{\gamma}+n^{\gamma}}. \end{array}\right. 因此当 r={\rm e}^{-2 \sum\limits_{1}^{n} j^{\gamma}+n^{\gamma}} 的时候, \log \mu(B(x, r)) / \log r 的最小值为 \frac{-\sum\limits_{1}^{n-1} j^{\gamma}}{-2 \sum\limits_{1}^{n} j^{\gamma}+n^{\gamma}}\approx \frac{-n^{\gamma+1} /(\gamma+1)}{-2 n^{\gamma+1} /(\gamma+1)+n^{\gamma}}=\frac{1}{2}-O(1 / n). 因此, \mu 的下局部维数在 B\left(\gamma, c_{1}(n), c_{2}(n), 1\right) 处处为 1/2,由 Frostman 引理 (见文献[2,定理 4.2]), 即 \dim_{\mathrm{H}} B\left(\gamma, c_{1}(n), c_{2}(n), 1\right) \geq \frac{1}{2}. 证毕. 定义新的集合 F_{\varphi}=\left\{x \in(0,1):\lim _{n \rightarrow \infty} \frac{L_{n}(x)}{\varphi(n)}=1\right\}. 其中函数 \varphi 是在引理 2.4 中定义的函数. 引理 3.2 对于前面定义的集合 F_{\varphi} (1) 当 \phi(n)=n^{\gamma} (\gamma>0) 时, 有 \dim_{\mathrm{H}} F_{\varphi}=1 ; (2) 当 \phi(n)={\rm e}^{n^{\gamma}} ( \gamma>0) 时, 有 \dim_{\mathrm{H}} F_{\varphi}=1, \gamma \in\left(0, \frac{1}{2}\right) ; \quad \dim_{\mathrm{H}} F_{\varphi} \geq \frac{1}{2}, \gamma \in\left[\frac{1}{2}, \infty\right). (1) 对于 \phi(n)=n^{\gamma} (\gamma>0) 的情况,注意到在文献[13]中证明了对任意的 \gamma \geq0 以及 \alpha \geq0,有 \dim_{\mathrm{H}}\left\{x \in(0,1): \lim _{n \rightarrow \infty} \frac{L_{n}(x)}{n^{\gamma}}=\alpha\right\}=1. 由其同样的方法, 容易得到对任意满足下式 \lim _{n \rightarrow \infty} \psi(n)=+\infty, \quad \lim _{n \rightarrow \infty} \frac{\log \psi(n)}{\log n}=\gamma<+\infty 的单调递增函数 \psi: \mathbb{N} \rightarrow \mathbb{R},集合 \left\{x \in(0,1): \lim _{n \rightarrow \infty} \frac{L_{n}(x)}{\psi(n)}=1\right\} 的 Hausdorff 维数是满的. 因此我们可以应用这个结果, 用引理 2.4 中定义的函数 \varphi 来代替 \psi , 然后我们得到 \dim_{\mathrm{H}} F_{\varphi}=1. (2) 对于 \phi(n)={\rm e}^{n^{\gamma}} 的情况. 首先我们讨论 \gamma\in (0,\frac{1}{2}) 的情形. 对任意的 \tau\in(0,1), 令 \delta\in(0,1) 为满足 (1+\frac{1}{1-\delta})(\frac{1}{2}-\tau)<1 的一个正实数. 对任意的 k\geq 1, 令 \varepsilon_k=\frac{1}{k^{\delta}}, 下面我们构造一列递增的整数序列 \{n_k\}_{k\geq1}, n_1 是使得对所有的 n\geq n_1$$\log\varphi(n)<n^{\frac{1}{2}-\tau}$$\varphi(n_1)\geq1 的最小整数. 对任意的 k\geq 2, 令 n_k=\min\left\{n\in \mathbb{Z}^+\colon \varphi(n)\geq (1+\varepsilon_{k-1})\varphi(n_{k-1}),\,\,n>n_{k-1}\right\}. M\geq 8 是一个正整数, 记 E_M(\varphi)=\{x\in(0,1)\colon d_{n_k}(x)=[(1+\varepsilon_k)\varphi(n_k)],\,\,\forall k\geq1,2 \leq d_i(x) \leq M, i\neq n_k \}. n_k 的定义, 容易验证 E_M(\varphi)\subset F_{\varphi}. 为了证明 \dim_{\mathrm{H}} E_M(\varphi)=1, 对任意的 \varepsilon>0, 我们将采用文献[13]的方法, 构造一个从 E_M(\varphi)$$E_M\frac{1}{1+\varepsilon}-Lipschitz 映射 f, 再由引理 2.2, 令 \varepsilon\to 0 以及M\to \infty, 则可得到 \dim_{\mathrm{H}} E_M(\varphi)=1. 对任意的 x\in E_M(\varphi), 令 y=f(x) 是将x的Lüroth展式中的所有 d_{n_k} 都去掉而得到的, 显然 y\in E_M.r(n):=\min\{k\colon n_k\leq n\}, 要验证映射 f 是一个 \frac{1}{1+\varepsilon}-Lipschitz 映射, 只需验证 \lim_{n\to\infty}\frac{r(n)}{n}=0 \lim_{n\to\infty}\frac{\log(d_{n_1}d_{n_2}\cdots d_{n_{r(n)}})}{n}=0. 对任意的 n\geq n_2,有 \begin{align*} \varphi(n) &\geq \varphi(n_{r(n)})\geq (1+\varepsilon_{r(n)-1})(1+\varepsilon_{r(n)-2})\cdots (1+\varepsilon_1)\varphi(n_1)\\ &\geq {\rm e}^{\frac{\varepsilon_1+\varepsilon_2+\cdots+\varepsilon_{r(n)-1}}{2}}\varphi(n_1). \end{align*} 则可知 \log \varphi(n)\geq \frac{(r(n)-1)^{1-\delta}-1}{2(1-\delta)}+\log\varphi(n_1),故而 r(n)^{1-\delta}\ll \log \varphi(n), 其中 Vinogradov 符号中的常数是绝对常数. 因为对所有的 n\geq n_1 都有 \log \varphi(n)<n^{\frac{1}{2}-\tau}, 我们有 r(n)\ll n^{\frac{\frac{1}{2}-\tau}{1-\delta}}. 再由 0<\tau<\frac{1}{2} 且满足(1+\frac{1}{1-\delta})(\frac{1}{2}-\tau)<1 可知,\frac{\frac{1}{2}-\tau}{1-\delta}<1.这意味着 (3.6) 式成立. \begin{align*}\log(d_{n_1}d_{n_2}\cdots d_{n_{r(n)}})&\leq \log((1+\varepsilon_1)(1+\varepsilon_2)\cdots (1+\varepsilon_{r(n)})\varphi(n_1)\cdots \varphi(n_{r(n)}))\\&\leq r(n)\log(\varphi(n))+\varepsilon_1+\cdots+\varepsilon_{r(n)}\\&\ll r(n)\log\varphi(n)+r(n)^{1-\delta}\ll r(n)n^{\frac{1}{2}-\tau}+r(n)^{1-\delta},\end{align*} 应用 (3.8) 式可知 (3.7) 式成立. 下面考虑 \gamma \in\left[\frac{1}{2}, \infty\right) 的情况.对于任意的 \varepsilon>0 , 存在 N \in \mathbb{N} 使得对于任意的 n \geq N 满足 \frac{1}{n}<\varepsilon . 对于 \gamma \in\left[\frac{1}{2}\right., \infty ), A(\gamma, \varphi)=\left\{x \in(0,1):\left(1-\frac{1}{n}\right) \varphi(n)<d_{n}(x)<\varphi(n), \forall n \geq N\right\}. 对于任意 x \in A(\gamma, \varphi) , 当 n 足够大的时候, 存在一个 1 \leq k \leq n , 使得 L_{n}(x)=d_{k}(x). 根据 \varphi 的单调性, 有 1-\varepsilon<1-\frac{1}{n}<\frac{d_{n}(x)}{\varphi(n)} \leq \frac{L_{n}(x)}{\varphi(n)} \leq \frac{d_{k}(x)}{\varphi(k)} <1+\varepsilon, 因此 A(\gamma, \varphi) \subset F_{\varphi}. c_{1}(n)=\left(1-\frac{1}{n}\right)\left(\frac{\beta+\alpha}{2}+\frac{\beta-\alpha}{2} \sin \left(\eta \cdot n^{\gamma}\right)\right), 以及 c_{2}(n)=\frac{\beta+\alpha}{2}+\frac{\beta-\alpha}{2} \sin \left(\eta \cdot n^{\gamma}\right). 很明显, c_{1}(n), c_{2}(n) 满足引理 3.1中的条件, 因此有 \dim_{\mathrm{H}} A(\gamma, \varphi)=\frac{1}{2}. 证毕. 最后, 由引理 2.4, 对于任意 x \in F_{\varphi} , 有 \begin{align*} \liminf _{n \rightarrow \infty} \frac{L_{n}(x)}{\phi(n)}=\lim _{n \rightarrow \infty} \frac{L_{n}(x)}{\varphi(n)}\cdot\liminf _{n \rightarrow \infty} \frac{\varphi(n)}{\phi(n)}=\alpha, \\ \limsup _{n \rightarrow \infty} \frac{L_{n}(x)}{\phi(n)}=\lim _{n \rightarrow \infty} \frac{L_{n}(x)}{\varphi(n)} \cdot \limsup _{n \rightarrow \infty} \frac{\varphi(n)}{\phi(n)}=\beta, \end{align*} 故可得 F_{\varphi} \subset F_{\phi}(\alpha, \beta). 因此, 根据引理 3.2, 当 \phi(n)=n^{\gamma}(\gamma>0) 或者 {\rm e}^{n^{\gamma}}\left(0<\gamma<\frac{1}{2}\right) 时, 有 \dim_{\mathrm{H}} F_{\phi}(\alpha, \beta)=1.\phi(n)={\rm e}^{n^{\gamma}}\left(\gamma \geq \frac{1}{2}\right) 时, 有 \dim_{\mathrm{H}} F_{\phi}(\alpha, \beta) \geq \frac{1}{2}. ### 3.2 当 \phi(n)={\rm e}^{n^{\gamma}}\left(\gamma>\frac{1}{2}\right) 时的上界 根据上面的下界估计, 我们只需讨论当 \phi(n)={\rm e}^{n^{\gamma}}\left(\gamma>\frac{1}{2}\right) 时集合 F_{\phi}(\alpha, \beta) 的上界. 对于任意 0<\alpha<\beta<\infty \gamma>\frac{1}{2}, 我们将证明 $\dim_{\mathrm{H}} F_{\phi}(\alpha, \beta) \leq \frac{1}{2}$. 在下文中, 为了方便, 我们将省略取整符号 $[\cdot]$, 结果不会受到影响.

$\dim_{\mathrm{H}} F_{\phi}(\alpha, \beta) \leq \frac{1}{2}.$

$(\alpha-\varepsilon) {\rm e}^{n_{k}^{\gamma}}\leq S_{n_{k}}(x) \leq n_{k}(\beta+\varepsilon) {\rm e}^{n_{k}^{\gamma}}$

$(\alpha-\varepsilon) {\rm e}^{n_{k-1}^{\gamma}}\leq S_{n_{k-1}}(x) \leq n_{k-1}(\beta+\varepsilon) {\rm e}^{n_{k-1}^{\gamma}}.$

$u_{k} \leq S_{n_{k}}(x)-S_{n_{k-1}}(x) \leq v_{k},$

$u_{k}=(\alpha-\varepsilon) {\rm e}^{k(\log k)^{\frac{1}{\gamma}}}-(\beta+\varepsilon)(k-1)^{\frac{1}{\gamma}}(\log (k-1))^{\frac{1}{{\gamma}^{2}}} {\rm e}^{(k-1)(\log (k-1))^{\frac{1}{\gamma}}}$

$v_{k}=(\beta+\varepsilon) k^{\frac{1}{\gamma}}(\log k)^{\frac{1}{\gamma^{2}}} {\rm e}^{k(\log k)^{\frac{1}{\gamma}}}.$

$F_{\phi}(\alpha, \beta) \subset \bigcup_{N=1}^{\infty} \bigcap_{k=N}^{\infty} \widetilde{B}(\gamma, N, k),$

$\widetilde{B}(\gamma, N, k):=\quad \bigcup_{\sum\limits_{i=n_{l-1}+1}^{n_{l}} d_{i} \in\left[u_{i}, \mathrm{v}_{i}\right], N \leq l \leq k} \quad I_{n_{k}}\left(d_{1}, \cdots, d_{n_{k}}\right).$

\begin{align*} \sum\limits_{I_{n_{k}} \subset \widetilde{B}(\gamma, N, k)}\left|I_{n_{k}}\right|^{s} & \leq \sum\limits_{I_{n_{k}} \subset \widetilde{B}(\gamma, N, k)} \prod_{l=N}^{k}\left(d_{n_{l-1}+1} \cdots d_{n_{l}}\right)^{-2 s} \\ & \leq \prod_{l=N}^{k} \sum\limits_{m \in\left[u_{l}, v_{l}\right]} \sum\limits_{\left(d_{n_{l-1}+1}, \cdots, d_{n_{l}}\right) \in B \left(m, n_{l}-n_{l-1}\right)}\left(d_{n_{l-1}+1} \cdots d_{n_{l}}\right)^{-2 s}\\ &\leq \prod_{l=N}^{k} u_{l}^{-2 s} v_{l}\left(\frac{25}{2}(2+\zeta(2 s))\right)^{n_{l}-n_{l-1}} \\ &=\prod_{l=N}^{k} C \cdot l^{\frac{1}{\gamma}} \cdot(\log l)^{\frac{1}{{\gamma}^{2}}} \cdot {\rm e}^{l(\log l)^{\frac{1}{\gamma}}(1-2 s)} \left(\frac{25}{2}(2+\zeta(2 s))\right)^{n_{l}-n_{l-1}}, \end{align*}

$n_{l}-n_{l-1}=l^{\frac{1}{\gamma}}(\log l)^{\frac{1}{{\gamma}^{2}}}-(l-1)^{\frac{1}{\gamma}}(\log (l-1))^{\frac{1}{{\gamma}^{2}}} \ll l^{\frac{1}{\gamma}-1}(\log l)^{\frac{1}{{\gamma}^{2}}}.$

$\dim_{\mathrm{H}} F_{\phi}(\alpha, \beta) \leq \frac{1}{2}, \,\,\,\, \phi(n)={\rm e}^{n^{\gamma}}\left(\frac{1}{2}<\gamma<1\right).$

$\left(\log k_{i-1}\right)^{2}<n_{k_{i}}-n_{k_{i-1}}<2\left(\log k_{i-1}\right)^{2}, \forall i \geq 2.$

$\dim_{\mathrm{H}} F_{\phi}(\alpha, \beta) \leq \frac{1}{2} \text, \,\,\,\, \phi(n)={\rm e}^{n^{\gamma}}(\gamma \geq 1).$

## 4 定理 1.4 的证明

$E_{\varphi}=\left\{x \in(0,1): \lim _{n \rightarrow \infty} \frac{S_{n}(x)}{\varphi(n)}=1\right\},$

$\dim_{\mathrm{H}} E_{\varphi}=1, \gamma \in\left(0, \frac{1}{2}\right) ; \quad \dim_{\mathrm{H}} E_{\varphi} \geq \frac{1}{2}, \gamma \in\left[\frac{1}{2}, \infty\right).$

$\gamma \in\left(0, \frac{1}{2}\right)$ 的时候,很容易验证

$\lim _{n \rightarrow \infty} \frac{\varphi(n)}{n} =\infty, \quad \limsup _{n \rightarrow \infty} \frac{\log \log \varphi(n)}{\log n}<\frac{1}{2},$

$\dim_{\mathrm{H}} E_{\varphi}=1, \quad 0<\gamma<\frac{1}{2}.$

$\gamma \geq \frac{1}{2}$ 时, 令

$\widetilde{E}(\gamma, \varphi)=\left\{x \in(0,1): \varphi(n)-\varphi(n-1)<d_{n}(x)<\left(1+\frac{1}{n}\right)(\varphi(n)-\varphi(n-1)), \forall n \geq N\right\},$

$\widetilde{E}(\gamma, \varphi) \subset E_{\varphi}.$

$c_{1}(n)=(\varphi(n)-\varphi(n-1)) \cdot {\rm e}^{-n^{\gamma}}, c_{2}(n)=c_{1}(n) \cdot\left(1+\frac{1}{n}\right).$

$\dim_{\mathrm{H}} \widetilde{E}(\gamma, \varphi)=\frac{1}{2},$

\begin{align*} &\liminf _{n \rightarrow \infty} \frac{S_{n}(x)}{\phi(n)}=\lim _{n \rightarrow \infty} \frac{S_{n}(x)}{\varphi(n)}\cdot\liminf _{n \rightarrow \infty} \frac{\varphi(n)}{\phi(n)}=\alpha, \\ &\limsup _{n \rightarrow \infty} \frac{S_{n}(x)}{\phi(n)} =\lim _{n \rightarrow \infty} \frac{S_{n}(x)}{\varphi(n)} \cdot \limsup _{n \rightarrow \infty} \frac{\varphi(n)}{\phi(n)}=\beta, \end{align*}

$E_{\varphi} \subset E(\gamma, \alpha, \beta).$

$x \in E(\gamma, \alpha, \beta)$, 对于任意 $0<\varepsilon<\alpha$, 取一个整数序列

$n_{k}=k^{\frac{1}{\gamma}}\left(\log \frac{\beta+\varepsilon}{\alpha-\varepsilon}+1\right)^{\frac{1}{\gamma}},\,\,\, k \geq 1.$

$(\alpha-\varepsilon) {\rm e}^{n_{k}^{\gamma}}<S_{n_{k}}(x)<(\beta+\varepsilon) {\rm e}^{n_{k}^{\gamma}}$

$(\alpha-\varepsilon) {\rm e}^{n_{k-1}^{\gamma}}<S_{n_{k-1}}(x)<(\beta+\varepsilon) {\rm e}^{n_{k-1}^{\gamma}}.$

$(\alpha-\varepsilon) {\rm e}^{n_{k}^{\gamma}}-(\beta+\varepsilon) {\rm e}^{n_{k-1}^{\gamma}} <S_{n_{k}}(x)-S_{n_{k-1}}(x) <(\beta+\varepsilon) {\rm e}^{n_{k}^{\gamma}}.$

\begin{align*} (\alpha-\varepsilon) {\rm e}^{n_{k}^{\gamma}}-(\beta+\varepsilon) {\rm e}^{n_{k-1}^{\gamma}} & ={\rm e}^{n_{k}^{\gamma}}\left((\alpha-\varepsilon)-(\beta+\varepsilon) {\rm e}^{n_{k-1}^{\gamma}-n_{k}^{\gamma}}\right) \\ & ={\rm e}^{n_{k}^{\gamma}}(\alpha-\varepsilon)\left(1-{\rm e}^{-1}\right). \end{align*}

$(\alpha-\varepsilon)\left(1-{\rm e}^{-1}\right) {\rm e}^{n_{k}^{\gamma}} <S_{n_{k}}(x)-S_{n_{k-1}}(x) <(\beta+\varepsilon) {\rm e}^{n_{k}^{\gamma}}.$

$\dim_{\mathrm{H}} E(\gamma, \alpha, \beta) \leq \frac{1}{2}, \quad \frac{1}{2}<\gamma<1.$

$k_{1}=3,\left(\log k_{i-1}\right)^{2}<n_{k_{i}}-n_{k_{i-1}}<2\left(\log k_{i-1}\right)^{2}, \forall i \geq 2.$

$\dim_{\mathrm{H}} E(\gamma, \alpha, \beta) \leq \frac{1}{2}, \quad \gamma \geq 1.$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Dajani K, Kraaikamp C.

Ergodic Theory of Numbers

Washington: Mathematical Association of America, 2002

Falconer K.

Fractal Geometry. Mathematical Foundations and Application

Chichester: John Wiley & Sons, 1990

Galambos J. Representations of Real Numbers by Infinite Series. Berlin: Springer, 1976

Galambos J.

Some remarks on the Lüroth expansion

Czechoslovak Math J, 1972, 22(97): 266-271

Galambos J.

The rate of the growth of the denominators in the Oppenheim series

Proc Amer Math Soc, 1976, 59(1): 9-13

Liao L M, Rams M.

Big Birkhoff sums in $d$-decaying Gauss like iterated function systems

Math Proc Cambridge Philos Soc, 2016, 160(3): 401-412

Lin S Y, Li J J, Lou M L.

Exceptional sets related to the largest digits in Lüroth expansions

Int J Number Theory, 2022, 18(7): 1429-1443

Lüroth J.

Ueber eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe

Math Ann, 1883, 21(3): 411-423

M Y.

The growth rate of the digits in the Lüroth expansions

Fractals, 2020, 28(4): 2050064

On the matrical theory of continued fraction mixing fibred systems and its application to Jacobi-Perron algorithm

Monatsh Math, 2003, 138(4): 267-288

Shen L M, Liu Y H.

A note on a problem of J. Galambos

Turkish J Math, 2008, 32(1): 103-109

Shen L M, Yu Y Y, Zhou Y X.

A note on the largest digits in Lüroth expansion

Int J Number Theory, 2014, 10(4): 1015-1023

Song K K, Fang L L, Ma J H.

Level sets of partial maximal digits for Lüroth expansion

Int J Number Theory, 2017, 13(10): 2777-2790

Tan X Y, Wang X J.

On the fast growth rate of the sum of digits of Lüroth expansion

Math Appl, 2018, 31(2): 300-304

Zhang M J, Ma C.

On the exceptional sets concerning the leading partial quotient in continued fractions

J Math Anal Appl, 2021, 500(1): 125100

Zhang M J, Wang W L.

On Lüroth expansions in which the largest digit grows with slowly increasing speed

Bull Aust Math Soc, 2023, 107(2): 204-214

/

 〈 〉