ON LÜROTH EXPANSIONS IN WHICH THE LARGEST DIGIT GROWS WITH SLOWLY INCREASING SPEED

Abstract

Let $0\leq \alpha \leq \infty $ , $0\leq a\leq b\leq \infty $ and $\psi $ be a positive function defined on $(0,\infty )$ . This paper is concerned with the growth of $L_{n}(x)$ , the largest digit of the first n terms in the Lüroth expansion of $x\in (0,1]$ . Under some suitable assumptions on the function $\psi $ , we completely determine the Hausdorff dimensions of the sets

$$\begin{align*}E_\psi(\alpha)=\bigg\{x\in(0,1]: \lim\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=\alpha\bigg\} \end{align*}$$

and

$$\begin{align*}E_\psi(a,b)=\bigg\{x\in(0,1]: \liminf\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=a, \limsup\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=b\bigg\}. \end{align*}$$

1 Introduction

It is well known that every $x\in (0,1]$ admits an infinite Lüroth expansion of the form

(1.1) $$ \begin{align} x=\frac{1}{d_1(x)}+\sum\limits_{n\geq2}\frac{1}{d_1(x)(d_1(x)-1)\cdots d_{n-1}(x)(d_{n-1}(x)-1)d_n(x)}, \end{align} $$

where $d_n(x)\in \mathbb {N}$ for all $n\geq 1$ , which we write as $x=[d_1(x),d_2(x),\ldots ]$ . Lüroth [12] showed that the Lüroth expansion can be induced by the Lüroth map $T:[0,1]\rightarrow [0,1]$ defined by

$$\begin{align*}T(x)= \begin{cases} 0 & \text{if }x=0, \\ \bigg\lfloor\dfrac{1}{x}\bigg\rfloor\bigg(\bigg(\bigg\lfloor\dfrac{1}{x}\bigg\rfloor+1\bigg)x-1\bigg) & \text{if }x\in(0,1]. \end{cases} \end{align*}$$

The digits $d_n:=d_n(x)$ in (1.1) are defined by

$$\begin{align*}d_1(x)=\bigg\lfloor\frac{1}{x}\bigg\rfloor+1 \quad\text{and}\quad d_n(x)=d_1(T^{n-1}(x)) \quad\text{for all } n\in\mathbb{N}, \end{align*}$$

where $\lfloor \cdot \rfloor $ denotes the integer part of some real number and $T^n$ stands for the nth iterate of $T\ (T^0=\mathrm {Id}_{(0,1]})$ .

Clearly, the above algorithm gives $d_n\geq 2$ for each $n\geq 1$ . Conversely, it is shown in [6] that any sequence of integers $\{d_n\}_{n\geq 1}$ with $d_n\geq 2$ for each $n\geq 1$ must be the Lüroth expansion of some $x\in (0,1]$ . The Lüroth expansion has been studied extensively in the representation theory of real numbers, probability theory and dynamical systems (see [1, 2, 5, 7] and the monograph of Dajani and Kraaikamp [3]).

Given $x\in (0,1]$ , let $L_n(x)=\max \{d_1(x),d_2(x),\ldots ,d_n(x)\}$ be the largest digit among the first n terms of the Lüroth expansion of x. The first metrical result on $L_n(x)$ was given by Galambos [6] in 1976: for Lebesgue almost all $x\in (0,1]$ ,

(1.2) $$ \begin{align} \lim\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log n}=1. \end{align} $$

That is, $\log L_n(x)$ tends to infinity steadily with the speed $\log n$ .

From the point of view of multifractal analysis, Shen et al. [14] studied the level sets

(1.3) $$ \begin{align} \bigg\{x\in(0,1]: \lim\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log n}=\gamma\bigg\}, \quad \gamma\geq0, \end{align} $$

and showed that they have full Hausdorff dimension. Recently, Lin and Li [11] generalised this result by considering the size of the sets for which the limit in (1.3) may not exist. More precisely, they proved that for $0\leq \alpha \leq \beta \leq \infty $ , the set

(1.4) $$ \begin{align} \bigg\{x\in(0,1]: \liminf\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log n}=\alpha, \limsup\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log n}=\beta\bigg\} \end{align} $$

has Hausdorff dimension one.

After (1.3) and (1.4), it is natural to wonder how large the sets are when $\log L_n(x)$ tends to infinity at a different rate. We will investigate the Hausdorff dimension of the sets when $\log L_n(x)$ grows with slowly increasing speed as defined below.

Definition 1.1 [8, 9].

Let $f(x)$ be a function defined on the interval $[c,\infty )$ such that $f(x)>0$ , $\lim _{x\rightarrow \infty }f(x)=\infty $ and with continuous derivative $f'(x)>0$ . We say the function $f(x)$ is slowly increasing if $ \lim _{x\rightarrow \infty }{xf'(x)}/{f(x)}=0. $

Slowly increasing functions were used recently by Jakimczuk [8, 9] as a tool to study the asymptotic properties of Bell numbers. Typical slowly increasing functions are $\log x$ , $\log \log x$ , $\log ^2x$ , ${\log x}/{\log \log x}$ . The elementary properties of slowly increasing functions will be presented in Section 2.

We complement the limit theorem (1.2) by studying the following two sets:

$$\begin{align*}E_\psi(\alpha):=\bigg\{x\in(0,1]: \lim\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=\alpha\bigg\}, \end{align*}$$

$$\begin{align*}E_\psi(a,b):=\bigg\{x\in(0,1]: \liminf\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=a, \limsup\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=b\bigg\}, \end{align*}$$

where $0\leq \alpha \leq \infty $ , $0\leq a\leq b\leq \infty $ and $\psi $ is a positive function defined on $(0,\infty )$ . We will establish the following two main theorems. We use $\dim _H$ to denote the Hausdorff dimension.

Theorem 1.2. If the function $\log \psi $ is slowly increasing, then $\dim _HE_\psi (\alpha )=1$ for any real number $\alpha $ with $0\leq \alpha \leq \infty $ .

Theorem 1.3. If the function $\log \psi $ is slowly increasing, then $\dim _HE_\psi (a,b)=1$ for any real numbers $a,b$ with $0\leq a\leq b\leq \infty $ .

In particular, we can take $\psi (x)=x^\gamma \ (\gamma>0)$ , $\psi (x)=x^{\log x}$ and $\psi (x)=\log x$ in Theorem 1.3 to give the following result.

Corollary 1.4. If $0\leq a\leq b\leq \infty $ and $\gamma>0$ , then

$$\begin{align*}\dim_HE_{\{n^\gamma\}}(a,b)=\dim_HE_{\{n^{\log n}\}}(a,b)=\dim_HE_{\{\log n\}}(a,b)=1. \end{align*}$$

Notice that if we take $\psi (n)=n$ in Theorems 1.2 and 1.3, then we obtain the special results $\dim _HE_{\psi }(\alpha )=\dim _HE_{\psi }(a,b)=1$ given in [11, 14]. Theorem 1.3 also implies the following result.

Corollary 1.5. If the function $\log \psi $ is slowly increasing, the set

$$\begin{align*}\bigg\{x\in(0,1]:\liminf\limits_{n\rightarrow\infty}\frac{\log L_{n}(x)}{\log\psi(n)}<\limsup\limits_{n\rightarrow\infty}\frac{\log L_{n}(x)}{\log\psi(n)} \bigg\} \end{align*}$$

has full Hausdorff dimension.

For more results concerning the largest digits in Lüroth expansions and continued fraction expansions, see [10, 15–17]. For the definitions and elementary properties of Hausdorff dimension, Falconer’s book [4] is recommended.

2 Preliminaries

In this section, we will list some elementary results related to Lüroth expansions and present some notation and basic facts that will be used later.

Let $\{d_n\}_{n\geq 1}$ be a sequence of integers not less than $2$ . We call

$$\begin{align*}I_n(d_1,\ldots,d_n)=\{x\in(0,1]: d_k(x)=d_k \text{ for } 1\leq k\leq n\} \end{align*}$$

a cylinder of level n, whose endpoints and length denoted by $|I_n(d_1,\ldots ,d_n)|$ are determined by the following lemma.

Lemma 2.1 [6].

Let $I_n(d_1,\ldots ,d_n)$ be a cylinder of level n. Then the left and right endpoints are

$$\begin{align*}\frac{1}{d_1}+\frac{1}{d_1(d_1-1)d_2}+\cdots+\prod\limits_{k=1}^{n-1}\frac{1}{d_k(d_k-1)}\frac{1}{d_n} \end{align*}$$

and

$$\begin{align*}\frac{1}{d_1}+\frac{1}{d_1(d_1-1)d_2}+\cdots+\prod\limits_{k=1}^{n-1}\frac{1}{d_k(d_k-1)}\frac{1}{d_n}+\prod\limits_{k=1}^{n}\frac{1}{d_k(d_k-1)}. \end{align*}$$

As a result,

$$\begin{align*}|I_n(d_1,\ldots,d_n)|=\prod\limits_{k=1}^{n}\frac{1}{d_k(d_k-1)}. \end{align*}$$

For $m\in \mathbb {N}$ with $m\geq 2$ , write $\Sigma _m=\{2,3,\ldots ,m\}$ . Let $E_m$ be the set consisting of all points in $(0,1]$ whose digits are less than m, that is,

$$\begin{align*}E_m=\{x\in(0,1]: d_n(x)\in\Sigma_m \text{ for all } n\geq1\}. \end{align*}$$

It is known that the set $E_m$ can be regarded as a self-similar set generated by contracting similarities $\{{x}/{a(a-1)}+{1}/{a}\}_{a=2}^{m}$ . The following lemma is a classic result which gives the dimension of $E_m$ .

Lemma 2.2 [7, 13].

For any $m\geq 2$ , $\dim _HE_m=s_m$ , where $s_m$ is the solution s of the equation

$$\begin{align*}\sum\limits_{2\leq a\leq m}\bigg(\frac{1}{a(a-1)}\bigg)^s=1. \end{align*}$$

Moreover, $\lim _{m\rightarrow \infty }s_m=1$ .

Next, we present a key tool which indicates that the Hausdorff dimensions of some specific sets are stationary to the dimension of $E_m$ under certain Hölder mappings defined below.

Let $\mathbb {J}=\{n_1<n_2<\cdots \}\subset \mathbb {N}$ and $f_{\mathbb {J}}: (0,1]\rightarrow (0,1]$ be a mapping satisfying

$$\begin{align*}f_{\mathbb{J}}: x=[d_1(x),d_2(x),\ldots]\mapsto \overline{x}:=\overline{[d_1(x),d_2(x),\ldots]}=[d_1(\overline{x}),d_2(\overline{x}),\ldots], \end{align*}$$

where the number $\overline {x}$ is obtained by deleting all $\{d_{n_k}(x)\}_{k\geq 1}$ in the Lüroth expansion of x. For $m\geq 2$ and $\{a_n\}_{n\geq 1}$ a sequence of integers, set

$$\begin{align*}F_m(\mathbb{J},\{a_k\}):=\{x\in(0,1]: d_{n_k}(x)=a_k, d_n(x)\in\Sigma_m \text{ for } n\neq n_k\text{ for all } k\geq1\}. \end{align*}$$

Lemma 2.3. Fix $m\geq 2$ and a set of positive integers $\mathbb {J}=\{n_1<n_2<\cdots \}$ . Let $\{a_k\}_{k\geq 1}$ be an increasing positive integer sequence satisfying $a_k\rightarrow \infty $ as $k\rightarrow \infty $ and

(2.1) $$ \begin{align} \lim\limits_{k\rightarrow\infty}\frac{k\log a_k}{n_k}=0. \end{align} $$

Then $ \dim _HF_m(\mathbb {J},\{a_k\})=\dim _Hf_{\mathbb {J}}(F_m(\mathbb {J},\{a_k\}))=\dim _HE_m=s_m. $

Proof. The main idea of the proof of Lemma 2.3 comes from [16]. Here we will modify the calculations in [14] and give a sketch of the proof of this argument.

To estimate the dimension of $F_m(\mathbb {J},\{a_k\})$ , we shall use the terminology of symbolic space. For each $n\geq 1$ , let

$$\begin{align*}D_n=\{(\sigma_1,\ldots,\sigma_n)\in\mathbb{N}^n: \sigma_{n_k}=a_k \text{ and } \sigma_i\in\Sigma_m, 1\leq i\neq n_k\leq n\}. \end{align*}$$

For any $n\geq 1$ and $(\sigma _1,\ldots ,\sigma _n)\in D_n$ , we call

$$\begin{align*}J_n(\sigma_1,\ldots,\sigma_n)=\bigcup\limits_{\sigma_{n+1}}I_{n+1}(\sigma_1,\ldots,\sigma_{n},\sigma_{n+1}) \end{align*}$$

the fundamental interval of level n, where the union is taken over all $\sigma _{n+1}$ such that $(\sigma _1,\ldots ,\sigma _{n},\sigma _{n+1})\in D_{n+1}$ . Clearly,

$$\begin{align*}F_m(\mathbb{J},\{a_k\})=\bigcap\limits_{n\geq1}\bigcup\limits_{(\sigma_1,\ldots,\sigma_{n})\in D_n}I_n(\sigma_1,\ldots,\sigma_{n}) =\bigcap\limits_{n\geq1}\bigcup\limits_{(\sigma_1,\ldots,\sigma_{n})\in D_n}J_n(\sigma_1,\ldots,\sigma_{n}). \end{align*}$$

By the definition of $f_{\mathbb {J}}$ with $\mathbb {J}=\{n_k\}_{k\geq 1}$ , we can assume that $n_k\leq n<n_{k+1}$ for some ${k\in \mathbb {N}}$ . Then $\overline {(\sigma _1,\ldots ,\sigma _{n})}:=f_{\mathbb {J}}((\sigma _1,\ldots ,\sigma _{n}))$ is obtained by deleting the k terms $\{\sigma _{n_i}\}_{i=1}^{k}$ in $(\sigma _1,\ldots ,\sigma _{n})$ . Write

$$\begin{align*}\overline{I_{n}}(\sigma_1,\ldots,\sigma_{n}):=I_{n-k}\overline{(\sigma_1,\ldots,\sigma_{n})}. \end{align*}$$

Then we have the following claim.

Claim 1. For any $\varepsilon>0$ , there exists $N_0>0$ such that for all $n\geq N_0$ and $(\sigma _1,\ldots ,\sigma _{n})\in D_n$ , we have

$$\begin{align*}|I_n(\sigma_1,\ldots,\sigma_{n})|\geq|\overline{I_{n}}(\sigma_1,\ldots,\sigma_{n})|^{1+\varepsilon}. \end{align*}$$

In fact, (2.1) implies that for any $\varepsilon>0$ , there exists $N_0>0$ such that for all $k>N_0$ , we have $k\log a_k<\tfrac {1}{2}\varepsilon \log 2 n_k$ . We can assume that $n_k\leq n<n_{k+1}$ and obtain

(2.2) $$ \begin{align} |I_n(\sigma_1,\ldots,\sigma_{n})|^{\varepsilon}\leq\frac{1}{2^{(n-k)\varepsilon}}\leq\frac{1}{2^{n_{k}\varepsilon}}\leq\frac{1}{a_{k}^{2k}}. \end{align} $$

Since $\{a_k\}$ is increasing, (2.2) and Lemma 2.1 give

$$ \begin{align*} |I_n(\sigma_1,\cdots,\sigma_{n})|&=|\overline{I_n}(\sigma_1,\ldots,\sigma_{n})| /{\sigma_{n_1}(\sigma_{n_1}-1)\cdots\sigma_{n_k}(\sigma_{n_k}-1)} \\ &\geq|\overline{I_n}(\sigma_1,\ldots,\sigma_{n})| /{a_k^{2k}} \\ &\geq|\overline{I_n}(\sigma_1,\ldots,\sigma_{n})|^{1+\varepsilon}. \end{align*} $$

Let x and y belong to the set $F_m(\mathbb {J},\{a_k\})$ with $x\neq y$ . It follows that there exists a largest integer n such that x and y are both contained in the same cylinder of level n. The next claim is devoted to estimating the distance between x and y, which is very similar to [14, Lemma 3.3], so we omit the details.

Claim 2. Let n be the largest level of the cylinders which contain both x and y. Then

$$\begin{align*}|y-x|\geq\min\bigg\{\frac{|I_n(\sigma_1,\ldots,\sigma_{n})|}{m^2\cdot a_n},\frac{|I_n(\sigma_1,\ldots,\sigma_{n})|}{m^3}\bigg\}. \end{align*}$$

Therefore, when $x,y\in F_m(\mathbb {J},\{a_k\})$ with

$$\begin{align*}|x-y|<\min\limits_{(\sigma_1,\ldots,\sigma_{N_0})\in D_{N_0}}\bigg\{\frac{I_{N_0}(\sigma_1,\ldots,\sigma_{N_0})}{m^2 a_{N_0+2}}, \frac{I_{N_0}(\sigma_1,\ldots,\sigma_{N_0})}{m^3}\bigg\}, \end{align*}$$

we have

$$\begin{align*}|f(x)-f(y)|\leq\max\{m^2 a_{N_0+2},m^3\}^{1+\varepsilon}\cdot|x-y|^{{1}/{\varepsilon}}. \end{align*}$$

From these two claims and [4, Proposition 2.3], we obtain

$$\begin{align*}\dim_HF_m(\mathbb{J},\{a_k\})\geq\frac{1}{1+\varepsilon}\dim_Hf_{\mathbb{J}}(F_m(\mathbb{J},\{a_k\}))=\frac{1}{1+\varepsilon}\dim_HE_m \end{align*}$$

and so $\dim _HF_m(\mathbb {J},\{a_k\})\geq \dim _Hf_{\mathbb {J}}(F_m(\mathbb {J},\{a_k\}))$ by letting $\varepsilon \rightarrow 0$ .

To see that $\dim _HF_m(\mathbb {J},\{a_k\})\leq \dim _Hf_{\mathbb {J}}(F_m(\mathbb {J},\{a_k\}))$ , it suffices to show that the mapping

$$\begin{align*}f_{\mathbb{J}}^{-1}: f_{\mathbb{J}}(F_m(\mathbb{J},\{a_k\}))\rightarrow F_m(\mathbb{J},\{a_k\}) \end{align*}$$

is $1$ -Hölder. For any $y_1,y_2\in f_{\mathbb {J}}(F_m(\mathbb {J},\{a_k\}))$ , let $y_1,y_2\in I_{n}(\sigma _1,\ldots ,\sigma _{n})$ with $\sigma _{n+1}(y_1)\neq \sigma _{n+1}(y_2)$ . Let $x_1=f_{\mathbb {J}}^{-1}(y_1),x_2=f_{\mathbb {J}}^{-1}(y_2)$ . By the definition of $f_{\mathbb {J}}^{-1}$ , we know that $x_1$ and $x_2$ are obtained by inserting the sequence $\{a_k\}_{k\geq 1}$ in the Lüroth expansions of $y_1$ and $y_2$ at the positions $\{n_{k}\}_{k\geq 1}$ , respectively. Let $M\in \mathbb {N}$ be such that we can insert just M integers $\{a_{i}\}_{i=1}^{M}$ into the block $(\sigma _1,\ldots ,\sigma _{n})$ . Then $x_1$ and $x_2$ have at least $n+M$ common digits in their Lüroth expansions. By Lemma 2.1,

$$ \begin{align*} \displaystyle |x_1-x_2|&\leq|I_{n+M}(\sigma_1,\ldots,\sigma_{n+M})| \\ \displaystyle &\leq|I_n(\sigma_1,\ldots,\sigma_{n})| /{\sigma_{n_1}(\sigma_{n_1}-1)\cdots\sigma_{n_k}(\sigma_{n_k}-1)} \\ \displaystyle &\leq\tfrac{1}{2}|I_n(\sigma_1,\ldots,\sigma_{n})|. \end{align*} $$

However, similar to the argument in Claim 2, we also have

$$\begin{align*}|y_1-y_2|\geq\min\bigg\{\frac{|I_n(\sigma_1,\ldots,\sigma_{n})|}{m^2\cdot a_n},\frac{|I_n(\sigma_1,\ldots,\sigma_{n})|}{m^3}\bigg\}. \end{align*}$$

It follows that

$$\begin{align*}|f_{\mathbb{J}}^{-1}(y_1)-f_{\mathbb{J}}^{-1}(y_2)|=|x_1-x_2|\leq\tfrac{1}{2}\max\{m^2 a_{N_0+2},m^3\}\cdot|y_1-y_2|, \end{align*}$$

showing that $f_{\mathbb {J}}^{-1}$ is $1$ -Hölder and $ \dim _HF_m(\mathbb {J},\{a_k\})\leq \dim _Hf_{\mathbb {J}}(F_m(\mathbb {J},\{a_k\})). $

We end this section by presenting the following lemma which exhibits some basic properties of slowly increasing functions.

Lemma 2.4 [8].

Let the functions $f(x)$ and $g(x)$ be slowly increasing and $\gamma $ be a positive constant. Then,

1. (1) the function $f(x^\gamma )$ is slowly increasing;

2. (2) the function $f(x^\gamma g(x))$ is slowly increasing;

3. (3) $\lim _{n\rightarrow \infty } {\log f(x)}/{\log x}=0;$

4. (4) $\lim _{n\rightarrow \infty }{f(x+1)}/{f(x)}=1.$

3 Proofs

This section is devoted to the proofs of our main results. To prove Theorem 1.2, we will construct a suitable subset $F_{m}(\mathbb {J},\{a_k\})$ of $E_\psi (\alpha )$ , so that the result can be established by using Lemma 2.3. As for the proof of Theorem 1.3, since the nonexistence of the limit in $E_\psi (a,b)$ describes the essence of the question compared with the known results, we need to carefully construct a nice Cantor subset in the lower bound estimations for the Hausdorff dimension. Our proof provides a convenient method to estimate the lower bound for the Hausdorff dimension, which is very different from the method used in [11].

Proof of Theorem 1.2.

The proof is divided into three cases according as $\alpha =0$ , $0<\alpha <\infty $ and $\alpha =\infty $ .

Case 1: $\alpha =0$ . In this case, it is clear that $E_m\subset E_\psi (0)$ . Therefore the result follows directly by Lemma 2.2.

Case 2: $0<\alpha <\infty $ . Let $m\geq 2$ and $\{a_n\}_{n\geq 1}$ be a sequence of integers and recall the set

$$\begin{align*}F_m(\mathbb{J},\{a_k\}):=\{x\in(0,1]: d_{n_k}(x)=a_k, d_n(x)\in\Sigma_m \text{ for } n\neq n_k \text{ for all } k\geq1\} \end{align*}$$

defined in Lemma 2.3. Here we take $n_k=k^2$ and $a_k=\lfloor \psi (k^2)^\alpha \rfloor $ for each $k\geq 1$ .

On the one hand, for any $x\in F_m(\mathbb {J},\{a_k\})$ , if $k^2\leq n<(k+1)^2$ for some integer k, then

$$\begin{align*}\frac{\log\lfloor\psi(k^2)^\alpha\rfloor}{\log\psi((k+1)^2)}\leq\frac{\log L_n(x)}{\log\psi(n)}\leq\frac{\log\lfloor\psi(k^2)^\alpha\rfloor}{\log\psi(k^2)}. \end{align*}$$

From Lemma 2.4(1) and (4),

$$\begin{align*}\lim\limits_{n\rightarrow\infty}\frac{\log\psi(k^2)}{\log\psi((k+1)^2)}=1. \end{align*}$$

Consequently,

$$\begin{align*}\lim\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=\alpha, \end{align*}$$

which yields $F_m(\mathbb {J},\{a_k\})\subset E_\psi (\alpha )$ .

On the other hand, since $\log \psi $ is slowly increasing, Lemma 2.4(3) implies that

$$\begin{align*}\lim\limits_{k\rightarrow\infty}\frac{\log\log\psi(k)}{\log k}=0, \end{align*}$$

which ensures that for any $\varepsilon $ with $0<\varepsilon <\tfrac {1}{2}$ and sufficiently large k,

(3.1) $$ \begin{align} \log\psi(k)<k^{\varepsilon}. \end{align} $$

This gives

$$\begin{align*}\lim\limits_{k\rightarrow\infty}\frac{k\log\lfloor\psi(k^2)^\alpha\rfloor}{k^2}\leq\lim\limits_{k\rightarrow\infty}\frac{\alpha\cdot k^{1+2\varepsilon}}{k^2}=0, \end{align*}$$

that is, (2.1) in Lemma 2.3 holds. From Lemma 2.3,

$$\begin{align*}\dim_{H}E_\psi(\alpha)\geq\dim_HF_m(\mathbb{J},\{a_k\})=\dim_HE_m=s_m \end{align*}$$

and we obtain the result in Theorem 1.2 by letting $m\rightarrow \infty $ .

Case 3: $\alpha =\infty $ . In this case, for each $k\geq 1$ , we take

$$\begin{align*}n_k=\lfloor k^2\log k\rfloor\quad \text{and} \quad a_k=\lfloor(\psi(k^2\log k))^{(\log k)^{{1}/{2}}}\rfloor \end{align*}$$

in the definition of the set $F_m(\mathbb {J},\{a_k\})$ in Lemma 2.3.

We show first that $F_m(\mathbb {J},\{a_k\})\subset E_\psi (\infty )$ . For every $x\in F_m(\mathbb {J},\{a_k\})$ , since the functions $\log \psi (x)$ and $\log x$ are slowly increasing, Lemma 2.4(2) and (4) give

$$\begin{align*}\lim\limits_{n\rightarrow\infty}\frac{\log(\psi(k^2\log k))}{\log(\psi((k+1)^2\log(k+1)))}=1. \end{align*}$$

So if $\lfloor k^2\log k\rfloor \leq n<\lfloor (k+1)^2\log (k+1)\rfloor $ for some integer k, then

$$\begin{align*}\lim\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}\geq\lim\limits_{k\rightarrow\infty}\frac{(\log k)^{1/2}\cdot\log(\lfloor\psi(k^2\log k)\rfloor)} {\log(\psi(\lfloor(k+1)^2\log(k+1)\rfloor))}=\lim\limits_{k\rightarrow\infty}(\log k)^{1/2}=\infty, \end{align*}$$

which means $F_m(\mathbb {J},\{a_k\})\subset E_\psi (\infty )$ .

Next, (3.1) holds for any $\varepsilon $ with $0<\varepsilon <\tfrac {1}{2}$ as in the last case, and we can check that (2.1) still holds here, namely

$$\begin{align*}\lim\limits_{k\rightarrow\infty}\frac{k\log\lfloor(\psi(k^2\log k))^{(\log k)^{{1}/{2}}}\rfloor}{\lfloor k^2\log k\rfloor} \leq\lim\limits_{k\rightarrow\infty}\frac{\log\psi(k^2\log k)}{k(\log k)^{{1}/{2}}}\leq\lim\limits_{k\rightarrow\infty}k^{2\varepsilon-1}(\log k)^{\varepsilon-{1}/{2}}=0. \end{align*}$$

Hence, by Lemma 2.3,

$$\begin{align*}\dim_{H}E_\psi(\infty)\geq\dim_HF_m(\mathbb{J},\{a_k\})=\dim_HE_m=s_m. \end{align*}$$

Then we finish the proof of Theorem 1.2 by letting $m\rightarrow \infty $ .

Proof of Theorem 1.3.

We give the proof of Theorem 1.3 for the case $0<a<b<\infty $ in detail. The argument for other cases involves minor modifications. In the following, we will write $\phi :=\log \psi $ for simplicity.

Case 1: $0<a<b<\infty $ . Let $\phi $ be a slowly increasing function. Our strategy is to find a nice Cantor subset of $E_\psi (a,b)$ with full Hausdorff dimension. To this end, we construct another slowly increasing function $\widetilde {\phi }$ satisfying some specific properties with respect to $\phi $ . Then the proof can be completed by using the result mentioned in Theorem 1.2.

For $0<a<b<\infty $ , define $\widetilde {\phi }(x)$ on $(0,\infty )$ such that $\widetilde {\phi }(x)>0$ and, for any $n\in \mathbb {N}$ ,

$$\begin{align*}\widetilde{\phi}(n)=\frac{a+b}{2}\phi(n)+\frac{b-a}{2}\phi(n)\sin\bigg(\frac{a}{b-a}\log\phi(n)\bigg). \end{align*}$$

Proposition 3.1. Let $\phi (n)$ be slowly increasing and define the function $\widetilde {\phi }$ as above. Then $\widetilde {\phi }$ is also slowly increasing and

(3.2) $$ \begin{align} \liminf\limits_{n\rightarrow\infty}\frac{\widetilde{\phi}(n)}{\phi(n)}=a,\quad \limsup\limits_{n\rightarrow\infty}\frac{\widetilde{\phi}(n)}{\phi(n)}=b. \end{align} $$

Proof. First, $0<a\cdot \phi \leq \widetilde {\phi }\leq b\cdot \phi $ and $\widetilde {\phi }\rightarrow \infty $ as $x\rightarrow \infty $ . Next, we check that the function $\widetilde {\phi }(x)$ has positive derivative. In fact,

$$ \begin{align*} \widetilde{\phi}'(x)&=\bigg(\frac{a+b}{2}\phi(x)+\frac{b-a}{2}\phi(x)\sin\bigg(\frac{a}{b-a}\log\phi(x)\bigg)\bigg)' \\ &=\frac{a+b}{2}\phi'(x)+\frac{b-a}{2}\phi'(x)\sin\bigg(\frac{a}{b-a}\log\phi(x)\bigg) \\ &\quad +\frac{b-a}{2}\phi(x)\cos\bigg(\frac{a}{b-a}\log\phi(x)\bigg)\frac{a}{b-a}\cdot\phi^{-1}(x)\cdot\phi'(x) \\ &\geq\frac{a+b}{2}\phi'(x)-\frac{b-a}{2}\phi'(x)-\frac{a}{2}\phi'(x) =\frac{a}{2}\phi'(x)>0, \end{align*} $$

where the last inequality follows from the fact that $\phi $ is slowly increasing. The calculation also implies that

$$\begin{align*}\bigg|\frac{x\widetilde{\phi}'(x)}{\widetilde{\phi}(x)}\bigg| \leq \bigg| \frac{x}{a\phi(x)} \bigg(\frac{a+b}{2}\phi'(x)+\frac{b-a}{2}\phi'(x)+\frac{a}{2}\phi'(x)\bigg) \bigg| \rightarrow0 \quad \text{as } x\rightarrow\infty. \end{align*}$$

Therefore, $\widetilde {\phi }$ is also a slowly increasing function. By the construction of $\widetilde {\phi }$ , (3.2) holds immediately.

Let $\widetilde {\phi }=\log \widetilde {\psi }$ be the slowly increasing function defined above, where $\widetilde {\psi }$ is a positive function defined on $(0,\infty )$ . We replace $\widetilde {\phi }$ with $\phi =\log \psi $ and take $\alpha =1$ in the set $E_\psi (\alpha )$ in Theorem 1.2. The Hausdorff dimension of the set

$$\begin{align*}E_{\widetilde{\psi}}(1):=\bigg\{x\in(0,1]: \lim\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\widetilde{\phi}(n)}=1\bigg\} \end{align*}$$

is full. The lower bound of $\dim _HE_\psi (a,b)$ follows directly by Proposition 3.1 and the fact that $E_{\widetilde {\psi }}(1)\subset E_\psi (a,b)$ . To see this, note that for any $x\in E_{\widetilde {\psi }}(1)$ ,

$$ \begin{align*} \liminf\limits_{n\rightarrow\infty}\frac{\log L_{n}(x)}{\phi(n)} &=\lim\limits_{n\rightarrow\infty}\frac{\log L_{n}(x)}{\widetilde{\phi}(n)}\cdot \liminf\limits_{n\rightarrow\infty}\frac{\widetilde{\phi}(n)}{\phi(n)}=a, \\ \limsup\limits_{n\rightarrow\infty}\frac{\log L_{n}(x)}{\phi(n)} &=\lim\limits_{n\rightarrow\infty}\frac{\log L_{n}(x)}{\widetilde{\phi}(n)}\cdot \limsup\limits_{n\rightarrow\infty}\frac{\widetilde{\phi}(n)}{\phi(n)}=b, \end{align*} $$

which means that $x\in E_\psi (a,b)$ .

Case 2: $0=a<b<\infty $ . The proof is similar to the case when $0<a<b<\infty $ . We only need to modify the construction of the function $\widetilde {\phi }$ to make sure that Proposition 3.1 still holds. We define $\widetilde {\phi }(x)$ on $(0,\infty )$ such that $\widetilde {\phi }(x)>0$ by taking

$$\begin{align*}\widetilde{\phi}(x)=\frac{b\phi(x)}{\log\log\phi(x)}+\frac{1}{2}b\phi(x)(\sin(\log\log\phi(x))+1). \end{align*}$$

Equation (3.2) holds directly and we can check that $\widetilde {\phi }(x)$ satisfies $\widetilde {\phi }'(x)>0$ and $|{x\widetilde {\phi }'(x)}/{\widetilde {\phi }(x)}|\rightarrow 0$ as $x\rightarrow \infty $ . Thus $\widetilde {\phi }(x)$ is slowly increasing.

For the remaining cases, the discussions run as before, so we only give the constructions of the slowly increasing functions $\widetilde {\phi }(x)$ as follows.

Case 3: $0<a<b=\infty $ . Take

$$\begin{align*}\widetilde{\phi}(x)=a\phi(x)+\phi(x)\log\phi(x)\bigg(\sin\bigg(\frac{a}{2}\log\log\phi(x)\bigg)+1\bigg). \end{align*}$$

Case 4: $0=a<b=\infty $ . Take

$$\begin{align*}\widetilde{\phi}(x)=\frac{\phi(x)}{\log\log\phi(x)}+\frac{1}{2}\phi(x)\log\phi(x)(\sin(\log\log\log\phi(x))+1). \end{align*}$$

Case 5: $0<a=b<\infty $ . Take $ \widetilde {\phi }(x)=a\phi (x). $

Case 6: $a=b=\infty $ . Take $ \widetilde {\phi }(x)=\phi (x)\log x. $

Case 7: $a=b=0$ . Take $ \widetilde {\phi }(x)=\log \phi (x). $