1 Introduction
Let
$(X, T, \mu )$
be a measure-theoretic dynamical system with a compatible metric
$d,$
where
$T\colon X\rightarrow X$
is a transformation on X and
$\mu $
is a finite T-invariant Borel measure. One of the most fundamental results in ergodic theory is the Poincaré recurrence theorem (see, for example, [Reference Furstenberg9]), which asserts that
$\mu $
-almost every point of X is recurrent in the sense that
$$ \begin{align*}\liminf_{n\to \infty}\,d\big(T^{n}x,x\big)=0.\end{align*} $$
That is to say, for
$\mu $
-almost every
$x\in X,$
the orbit
$\{T^{n}x\}_{n\geq 0}$
returns to shrinking neighbourhoods of the initial point x infinitely often. However, this result gives no information about the speed at which a generic orbit
$\{T^{n}x\}_{n\geq 0}$
comes back to the starting point. This motivates the study of quantitative aspects of the distribution of T-orbits in X, leading to areas such as the dynamical Borel–Cantelli lemma [Reference Chernov and Kleinbock5], shrinking target problems [Reference Hill and Velani13, Reference Li, Liao, Velani and Zorin14], hitting time [Reference Galatolo10] and logarithm-type laws [Reference Galatolo and Peterlongo11, Reference Pawelec and Urbański16]. A fundamental quantity in this study is the shortest distance between the orbit of a point x and a fixed point
$y_{0}\colon $
$$ \begin{align*}d_{n}(x,y_{0})=\min_{0\leq i\leq n-1}\,d\big(T^{i}x, y_{0}\big).\end{align*} $$
Galatolo and Peterlongo [Reference Galatolo and Peterlongo11] analysed the asymptotic behaviour of
$d_{n}(x,y_{0})$
and established its close relationship with hitting times.
A natural generalisation of this problem is to study the proximity between two entire orbits rather than an orbit and a fixed point. Barros et al. [Reference Barros, Liao and Rousseau2] defined the shortest distance between the orbits of two points x and y,
$$ \begin{align*}m_{n}(x,y):=\min_{0\leq i,j\leq n-1}\,d\big(T^{i}x, T^{j}y\big),\end{align*} $$
and proved that its asymptotic behaviour is governed by the correlation dimension of the invariant measure
$\mu ,$
defined as the limit (when it exists)
$$ \begin{align*} C_{\mu}=\lim_{r\rightarrow 0}\frac{\log \int_{X}\mu(B(x,r))\,d\mu(x)}{\log r}. \end{align*} $$
More precisely, if the correlation dimension
$C_{\mu }$
exists, then under some rapid mixing conditions of the system, they proved that for
$\mu \otimes \mu $
-almost all
$(x,y)\in X\times X,$
$$ \begin{align} \lim_{n\rightarrow\infty}\frac{-\log m_{n}(x,y)}{\log n}=\frac{2}{C_{\mu}}. \end{align} $$
Closely related problems have aroused great interest in recent years: for orbits encoded by a measurable function (see [Reference Coutinho, Lambert and Rousseau6]); for n-orbits of a typical collection of k points (see [Reference Barros and Rousseau3]); for random dynamical systems (see [Reference Gouëzel, Rousseau and Stadlbauer12, Reference Rousseau17]); for one-dimensional Gibbs–Markov interval maps (see [Reference Zhao22]). More detailed presentations of related problems and their applications can be found in [Reference Caby4, Reference Fan, Liao and Qiu8, Reference Shi and Zhou19, Reference Todd and Rousseau20].
After the work of Barros et al. [Reference Barros, Liao and Rousseau2], we are interested in the set of points violating the limit in (1.1). Those sets usually have zero measure in probability and dynamics, but they are very often studied in fractal geometry because they can be characterised by fractal tools.
To begin with, let
$\Phi =\{\phi _{i}\colon [0,1]\rightarrow [0,1], \ i\in \Lambda \}$
be a collection of injective mappings of
$[0,1]$
with
$\Lambda $
a countable index set. We call
$\Phi $
an iterated function system (IFS) if there exists
$\rho $
with
$0<\rho <1$
such that
$$ \begin{align*} |\phi_{i}(x)-\phi_{i}(y)|\leq \rho |x-y| \end{align*} $$
for every
$i\in \Lambda $
and for every pair of points
$x,y\in X.$
Definition 1.1. We call the IFS
$\Phi =\{\phi _{i}\colon [0,1]\rightarrow [0,1], \ i\in \Lambda \}$
a conformal IFS if the following conditions are satisfied.
-
(1) Open set condition (OSC):
$\phi _{i}((0,1))\subset (0,1)$
for
$i\in \Lambda $
, and
$$ \begin{align*}\phi_{i}((0,1))\cap\phi_{j}((0,1))=\emptyset \quad\mbox{for } i,j\in \Lambda, i\neq j.\end{align*} $$
-
(2) There exists an open connected set
$V\subset \mathbb {R}$
such that
$[0,1]\subset V$
and all maps
$\phi _{i}$
extend to
$C^{1}$
-diffeomorphisms of V into
$V.$
-
(3) Bounded distortion property (BDP): there exists
$K\geq 1$
such that for any
$x,y\in V$
and
$(\omega _{1}, \omega _{2},\ldots ,\omega _{n})\in \Lambda ^{n}$
with
$n\geq 1$
,
$$ \begin{align*} K^{-1}\leq \frac{|(\phi_{\omega_{1}}\circ\phi_{\omega_{2}}\circ\cdots\circ\phi_{\omega_{n}})'(x)|}{|(\phi_{\omega_{1}}\circ\phi_{\omega_{2}}\circ\cdots\circ\phi_{\omega_{n}})'(y)|}\leq K. \end{align*} $$
The attractor J associated with a conformal IFS
$\Phi $
can be defined as the image of the symbolic space under a projection map. Let
$\Lambda ^{\ast }=\bigcup _{n\geq 1}\Lambda ^{n}$
and
$\Lambda ^{\mathbb {N}}$
be the space of finite words and infinite words over
$\Lambda $
, respectively. For
$\omega =(\omega _{1}, \omega _{2},\ldots ,\omega _{n})\in \Lambda ^{n},$
let
$\phi _{\omega }=\phi _{\omega _{1}}\circ \phi _{\omega _{2}}\circ \cdots \circ \phi _{\omega _{n}}.$
Given
$\omega \in \Lambda ^{\ast }\cup \Lambda ^{\mathbb {N}},$
if the length
$|\omega |$
of
$\omega $
is greater than the integer
$n\geq 1,$
we denote by
$\omega |_{n}$
the word
$(\omega _{1},\omega _{2},\ldots ,\omega _{n}).$
For each
$\omega \in \Lambda ^{\mathbb {N}},$
the sequence of compact sets
$\{\phi _{\omega |_n}([0,1])\}_{n\geq 1}$
is nested. Thus, the set
$\bigcap _{n\geq 1}\phi _{\omega |_n}([0,1])$
is a singleton, which we denote by
$\pi (\omega ).$
Therefore,
$$ \begin{align*}J=\pi(\Lambda^{\mathbb{N}})=\bigcup_{\omega\in\Lambda^{\infty}}\bigcap_{n\geq1}\phi_{\omega|_n}([0,1])=\bigcap_{n\geq1}\bigcup_{\omega\in\Lambda^{n}}\phi_{\omega}([0,1]).\end{align*} $$
We call the map
$\pi \colon \Lambda ^{\mathbb {N}}\rightarrow J$
a projection map and
$\omega \in \Lambda ^{\mathbb {N}}$
a coding of
$x\in J$
if
$\pi (\omega )=x.$
By the open set condition in Definition 1.1, there are at most countably many points
$x\in J$
having more than one digit sequence. Since a countable set is negligible in the sense of the Hausdorff dimension, we may assume that every point
$x\in J$
has a unique digit sequence, denoted by
$\{\omega _{n}(x)\}_{n\geq 1}$
, and we write
$$ \begin{align*}x=[\omega_{1}(x),\omega_{2}(x),\ldots].\end{align*} $$
For
$n\ge 1$
and
$(\omega _1,\ldots ,\omega _n)\in \Lambda ^{n},$
the set
$$ \begin{align*}J_{n}(\omega_1,\ldots,\omega_n) =\phi_{\omega_{1}}\circ\cdots\circ\phi_{\omega_{n}}([0,1])\end{align*} $$
is called an nth-order basic cylinder. Except for a countable set, the nth-order cylinder
$J_{n}(\omega _1,\ldots ,\omega _n)$
is identical with the set of points
$x\in J$
whose digit sequences begin with
$\omega _1,\ldots ,\omega _n.$
Now, we define the map
$T\colon J\rightarrow J$
as follows: for each
$x\in J,$
$$ \begin{align*}Tx=T([\omega_{1}(x),\omega_{2}(x),\omega_3(x),\ldots])=[\omega_{2}(x),\omega_{3}(x),\ldots].\end{align*} $$
In the system
$(J,T),$
the shortest distance
$m_{n}(x,y)$
for orbits can be rewritten as
$$ \begin{align*}m_{n}(x,y):=\min_{0\leq i,j\leq n-1}d\big(T^{i}x, T^{j}y\big),\end{align*} $$
where
$x,y\in J$
and
$d(x,y)=\exp \{-\inf \{i\geq 1\colon \omega _{i}(x)\neq \omega _{i}(y)\}\}.$
We use this symbolic metric to encode common prefix lengths because of our problem’s inherent connection to sequence matching. For
$0\leq \alpha \leq \beta \leq \infty ,$
we define the exceptional sets
$$ \begin{align*}E_{\alpha,\beta}=\bigg\{(x,y)\in J\times J\colon \liminf_{n\rightarrow\infty}\frac{-\log m_{n}(x,y)}{\log n}=\alpha, \limsup_{n\rightarrow\infty}\frac{-\log m_{n}(x,y)}{\log n}=\beta\bigg\}.\end{align*} $$
The following result shows that the exceptional sets are large from the dimensional point of view.
Theorem 1.2. Let
$\Phi =\{\phi _{i}\colon [0,1]\rightarrow [0,1], \ i\in \Lambda \}$
be a conformal IFS such that all the derivatives
$\{\phi ^{\prime }_{n}(x)\}$
have the same sign. For
$0\leq \alpha \leq \beta \leq \infty $
:
-
(1)
$2\dim _{H}J\leq \dim _{H}E_{\alpha ,\beta }\leq \dim _{H}J+\dim _{P}J$
; -
(2) if
$\Lambda $
is finite, then
$\dim _{H}E_{\alpha ,\beta }=2\dim _{H}J$
; -
(3)
$E_{\alpha ,\beta }$
is dense in
$J\times J$
.
Throughout this paper,
$\dim _H$
and
$\dim _{P}$
denote the Hausdorff dimension and the packing dimension, respectively. The condition that all derivatives
$\{\phi _n'(x)\}$
have the same sign ensures uniform contraction orientation, which is used in the proof to control distortion in Cantor set constructions.
This paper was also motivated by a result of Arratia and Waterman [Reference Arratia and Waterman1] on the longest common substring problem for random sequences. More precisely, for two sequences X and
$Y,$
the length of the longest common substring, allowing shifts, is
$$ \begin{align*}M_{n}(X,Y):=\max\big\{k\in \mathbb{N}\colon X_{i+1}=Y_{j+1},\ldots,X_{i+k}=Y_{j+k} ~\text{for some}~ i,j, 0\leq i,j \leq n-k\big\}.\end{align*} $$
Arratia and Waterman showed that if
$X_{1},X_{2},\ldots ,Y_{1},Y_{2},\ldots $
are independent and identically distributed, then
$$ \begin{align*}\mathbb{P}\,\bigg(\lim_{n\rightarrow\infty}\frac{M_{n}}{\log n}=\frac{2}{\log 1/p}\bigg)=1,\end{align*} $$
where
$p=\mathbb {P}(X_{1}=Y_{1})$
is the collision probability. Furthermore, they also proved that the same result holds for Markov chains, but with p being the largest eigenvalue of the matrix
$[(p_{ij})^{2}],$
where
$[p_{ij}]$
is the transition matrix. This result was recently generalised in [Reference Barros, Liao and Rousseau2] for
$\alpha $
-mixing processes with exponential decay and
$\psi $
-mixing processes with polynomial decay with a limit depending on the Rényi entropy of
$\mathbb {P}.$
In fact, they proved that if the Rényi entropy
$H_{2}$
exists, then
$$ \begin{align} \mathbb{P}\,\bigg(\lim_{n\rightarrow\infty}\frac{M_{n}}{\log n}=\frac{2}{H_{2}}\bigg)=1. \end{align} $$
These results reveal the behaviour of the shortest distance between orbits from the perspective of dynamical systems. In this paper, we consider this problem from the point of view of fractal dimension.
Let us now turn to consider the set of points violating the limit in (1.2). For
$x=[\omega _{1}(x),\omega _{2}(x),\ldots ],\ y=[\omega _{1}(y),\omega _{2}(y),\ldots ]\in J,$
the length of the longest matching consecutive subsequence
$M_{n}(x,y)$
of x and y in the first n digits is defined by
$$ \begin{align*} M_{n}(x,y):=\max\bigg\{k\in \mathbb{N}\colon \begin{array}{c@{}} \omega_{i+1}(x)=\omega_{j+1}(y),\ldots, \omega_{i+k}(x)=\omega_{j+k}(y)\\ \mbox{for some } i,j \mbox{ with } 0\leq i,j \leq n-k \end{array} \bigg\}. \end{align*} $$
For
$0\leq \alpha \leq \beta \leq \infty ,$
we define the exceptional sets
$$ \begin{align*}F_{\alpha,\beta}=\bigg\{(x,y)\in J\times J\colon \liminf_{n\rightarrow\infty}\frac{M_{n}(x,y)}{\log n}=\alpha, \limsup_{n\rightarrow\infty}\frac{M_{n}(x,y)}{\log n}=\beta\bigg\}. \end{align*} $$
Theorem 1.3. Let
$\Phi =\{\phi _{i}\colon [0,1]\rightarrow [0,1], \ i\in \Lambda \}$
be a conformal IFS such that all the derivatives
$\{\phi ^{\prime }_{n}(x)\}$
have the same sign. For
$0\leq \alpha \leq \beta \leq \infty $
:
-
(1)
$2\dim _{H}J\leq \dim _{H}F_{\alpha ,\beta }\leq \dim _{H}J+\dim _{P}J$
; -
(2) if
$\Lambda $
is finite, then
$\dim _{H}F_{\alpha ,\beta }=2\dim _{H}J$
; -
(3)
$F_{\alpha ,\beta }$
is dense in
$J\times J$
.
2 Preliminaries
This section is devoted to introducing some properties for conformal IFSs. For more information on conformal IFSs, we refer to [Reference Mauldin and Urbański15]. We also cite some basic techniques for estimating the Hausdorff dimension of a fractal set (see [Reference Falconer7]).
Let
$\Phi =\{\phi _{i}\colon [0,1]\rightarrow [0,1], \ i\in \Lambda \}$
be a conformal IFS with the attractor
$J.$
Let
$M>1$
be a positive integer and
$$ \begin{align*}J_{M}=\{x\in J\colon 1\leq \omega_{n}(x)\leq M, \ n\geq1\}.\end{align*} $$
Then,
$J_{M}$
is the attractor of the finite IFS,
$\{\phi _{i}\colon [0,1]\rightarrow [0,1], \ 1\leq i\leq M\}$
.
Lemma 2.1 [Reference Mauldin and Urbański15, Theorem 3.15].
We have
$\lim _{M\rightarrow \infty }\dim _{H}J_{M}=\dim _{H}J.$
For
$1\leq c\leq M, p>1$
, set
$$ \begin{align*}J_{M,p,c}=\{x\in J_{M}\colon \omega_{kp+1}(x)=c, \ k\geq0\}.\end{align*} $$
The Hausdorff dimension of
$J_M$
can be approximated by that of
$J_{M,p,c}$
.
Lemma 2.2 [Reference Shi, Tan and Zhou18, Lemma 4.1].
For
$\epsilon>0, 1\leq c\leq M,$
there exists a positive integer
$p_{0}$
such that
$\dim _{H}J_{M,p,c}\geq \dim _{H}J_{M}-\epsilon $
for all
$p\geq p_{0}.$
Let
$\{k_{n}\}_{n\geq 1}$
be an infinite subsequence of
$\mathbb {N}$
and let
$\{B_{n}\}_{n\geq 1}$
be a sequence of finite blocks over
$\{1,2,\ldots , M\}$
, say
$B_{n}=(a^n_{1},\ldots ,a^n_{b_{n}})$
. For
$x\in J_{M,p,c},$
we write its coding as
$$ \begin{align*}(\omega_{1},\ldots,\omega_{k_{1}},\omega_{k_{1}+1},\ldots,\omega_{k_{2}},\omega_{k_{2}+1},\ldots).\end{align*} $$
We insert the blocks
$\{B_{n}\}_{n\geq 1}$
after the positions
$\{k_n\}_{n\ge 1}$
in the coding of x and obtain a new sequence
$$ \begin{align*}(\omega_{1},\ldots,\omega_{k_{1}}, a^1_{1},\ldots,a^1_{b_{1}} ,\omega_{k_{1}+1},\ldots,\omega_{k_{2}},a^2_{1},\ldots,a^2_{b_{2}} ,\omega_{k_{2}+1},\ldots).\end{align*} $$
There exists a unique point
$y\in J$
which possesses the above sequence as its coding. Denote the collection of those y by
$J_{M,p,c}\big (\{k_{n}\},\{B_{n}\}\big ).$
Let
$b_n=|B_n|.$
Lemma 2.3 [Reference Shi, Tan and Zhou18, Lemma 4.2].
Let
$\Phi =\{\phi _{i}\colon [0,1]\rightarrow [0,1], \ i\in \Lambda \}$
be a conformal IFS with the property that all the derivatives
$\{\phi ^{\prime }_{n}(x)\}$
have the same sign. If
$\lim _{n\rightarrow \infty }{(b_{1}+b_{2}+\cdots +b_{n})}/{k_{n}}=0,$
then
$$ \begin{align*}\dim_{H}J_{M,p,c}=\dim_{H}J_{M,p,c}\big(\{k_{n}\},\{B_{n}\}\big).\end{align*} $$
The following product formula is cited here for later use.
Lemma 2.4 [Reference Wang and Wu21, Theorem 1.2].
If
$E\subset \mathbb {R}^{d}, \ F\subset \mathbb {R}^{n},$
then
$$ \begin{align*}\dim_{H}E+\dim_{H}F\leq\dim_{H}(E\times F)\leq\dim_{H}E+\dim_{P}F.\end{align*} $$
3 Proof of Theorem 1.2
This section is devoted to proving Theorem 1.2. From now on,
$\lfloor x\rfloor $
denotes the integer part of the real number
$x.$
3.1 Estimation of
$\mathbf{dim} _{H}\ E_{\alpha ,\beta }$
We divide the whole proof into two parts: a detailed proof for the case
$0<\alpha < \beta <+\infty $
and sketches of the proof for the remaining cases.
Case 1:
$0<\alpha <\beta <\infty $
. Assume without loss of generality that
$\beta \log n\geq 1$
for all
${n\geq 1.}$
We choose two integer sequences
$\{n_{k}\}_{k\geq 0}$
and
$\{c_{k}\}_{k\geq 0}$
satisfying
$$ \begin{align*}n_{0}>1,\quad e^{({\beta}/{\alpha})^{k}}-1\leq n_{k}<e^{({\beta}/{\alpha})^{k}},\quad c_{k}=\lfloor\beta\log n_{k}\rfloor.\end{align*} $$
It is clear that
$$ \begin{align} \lim_{k\rightarrow\infty}\frac{\log n_{k+1}}{\log n_{k}}=\frac{\beta}{\alpha}, \end{align} $$
and so
$\{n_{k}\}$
increases super-exponentially. Note that
$$ \begin{align*}\log(n_{k}+c_{k})= \log n_{k} +\log \bigg(1+\frac{c_{k}}{n_{k}}\bigg)\leq \log n_{k}+\frac{c_{k}}{n_{k}}.\end{align*} $$
So, we obtain
$$ \begin{align} \lim_{k\rightarrow\infty}\frac{\log(n_{k}+c_{k})}{\log n_{k}}=1. \end{align} $$
By the definitions of the sequences
$\{n_{k}\}_{k\geq 0}$
and
$\{c_{k}\}_{k\geq 0},$
$$ \begin{align} \lim_{k\rightarrow\infty}\frac{c_{k}}{n_{k+1}-n_{k}}=0. \end{align} $$
Thus, we can assume without loss of generality that
$n_{k+1}-n_{k}>c_{k}$
for
$k\geq 0.$
Now, we construct a Cantor-type subset of
$E_{\alpha ,\beta }.$
For
$k\geq 0,$
put
$$ \begin{align*}n_{k+1}-n_{k}=c_{k}\cdot\iota_{k}+\theta_{k},\end{align*} $$
where
$$ \begin{align*}\iota_{k}=\bigg\lfloor\frac{n_{k+1}-n_{k}}{c_{k}}\bigg\rfloor, \quad 0\leq\theta_{k}<c_{k}.\end{align*} $$
Define the marked set D by
$$ \begin{align*}D=\bigcup_{k\ge0}\{n_{k},n_{k}+1,n_{k}+2,\ldots,n_{k}+c_{k}, n_{k}+2c_{k},n_{k}+3c_{k},\ldots, n_{k}+\iota_kc_{k}\}.\end{align*} $$
Set
$a_{n}=1$
for
$n\in D$
and
$$ \begin{align*}b_{n_{k}}=M, \quad b_{n_{k}+1}=\cdots=b_{n_{k}+c_{k}-1}=1, \quad b_{n_{k}+c_{k}}=b_{n_{k}+2c_{k}}=\cdots=b_{n_{k}+\iota_kc_{k}}=M.\end{align*} $$
For
$n\geq 1,$
we define the sets
$$ \begin{align*}D_{n}&=\big\{(\varepsilon_{1},\ldots,\varepsilon_{n}) \in\{1,2,\ldots,M-1\}^n\colon \varepsilon_{k}=a_{k}\text{ if }k\in D\big\},\\ D^{\prime}_{n}&=\big\{(\varepsilon_{1},\ldots,\varepsilon_{n}) \in\{1,2,\ldots,M\}^n\colon \varepsilon_{k}=b_{k}\text{ if }k\in D\big\}.\end{align*} $$
The Cantor-type subsets are defined by
$$ \begin{align*}E_{M}(\{n_k\}, \{c_k\})=\bigcap_{n\geq1}\bigcup_{(\varepsilon_{1},\ldots,\varepsilon_{n})\in D_{n}}J_{n}(\varepsilon_{1},\ldots,\varepsilon_{n}),\end{align*} $$
$$ \begin{align*}E^{\prime}_{M}(\{n_k\}, \{c_k\})=\bigcap_{n\geq1}\bigcup_{(\varepsilon^{\prime}_{1},\ldots,\varepsilon^{\prime}_{n})\in D^{\prime}_{n}}J_{n}(\varepsilon^{\prime}_{1},\ldots,\varepsilon^{\prime}_{n}).\end{align*} $$
We make the following claims.
Claim 1.
$E_{M}(\{n_k\}, \{m_k\})\times E^{\prime }_{M}(\{n_k\}, \{c_k\})\subset E_{\alpha ,\beta }$
.
Fix a point
$(x,y)\in E_{M}(\{n_k\}, \{c_k\})\times E^{\prime }_{M}(\{n_k\}, \{c_k\}).$
For sufficiently large
$n\in \mathbb {N},$
there exists
$k\in \mathbb {N}$
such that
$n_{k}\leq n< n_{k+1}:$
-
(i) if
$n_{k}\leq n<n_{k}+c_{k},$
then
$e^{-c_{k}}\leq m_{n}(x,y)\leq e^{-c_{k-1}}$
; -
(ii) if
$n_{k}+c_{k}\leq n<n_{k+1},$
then
$m_{n}(x,y)=e^{-c_{k}}$
.
Combining these inequalities with (3.1) and (3.2), we deduce that
$$ \begin{align*} \liminf_{n\rightarrow\infty}\frac{-\log m_{n}(x,y)}{\log n}\geq \min\bigg\{\lim_{k\rightarrow\infty}\frac{c_{k-1}}{\log (n_{k}+c_{k})},\lim_{k\rightarrow\infty}\frac{c_{k}}{\log n_{k+1}}\bigg\} =\alpha \end{align*} $$
and
$$ \begin{align*} \limsup_{n\rightarrow\infty}\frac{-\log m_{n}(x,y)}{\log n}\leq \max\bigg\{\lim_{k\rightarrow\infty}\frac{c_{k}}{\log n_{k}},\lim_{k\rightarrow\infty}\frac{c_{k}}{\log(n_{k}+c_{k})}\bigg\} =\beta. \end{align*} $$
Note that
$m_{n_{k+1}}(x,y)=m_{n_{k}+c_{k}}(x,y)=e^{-c_{k}}.$
By (3.1), (3.2) again, it follows that
$$ \begin{align*} \lim_{k\rightarrow\infty}\frac{-\log m_{n_{k+1}}(x,y)}{\log n_{k+1}}=\lim_{k\rightarrow\infty}\frac{c_{k}}{\log n_{k+1}}=\alpha\end{align*} $$
and
$$ \begin{align*}\lim_{k\rightarrow\infty}\frac{-\log m_{n_{k}+c_{k}}(x,y)}{\log(n_{k}+c_{k})}=\lim_{k\rightarrow\infty}\frac{c_{k}}{\log(n_{k}+c_{k})}=\beta.\end{align*} $$
Thus,
$$ \begin{align*}\liminf_{n\rightarrow\infty}\frac{-\log m_{n}(x,y)}{\log n}=\alpha \quad\text{and}\quad \limsup_{n\rightarrow\infty}\frac{-\log m_{n}(x,y)}{\log n}=\beta,\end{align*} $$
which implies Claim 1.
Claim 2. The density of
$D\subset \mathbb {N}$
is zero.
For sufficiently large
$n\in \mathbb {N},$
there exists
$k\in \mathbb {N}$
such that
$n_{k}\leq n< n_{k+1}.$
-
(i) If
$n_{k}\leq n\leq n_{k}+c_{k},$
then
$\sharp \{i\leq n,i\in D\}=\sum _{j=0}^{k-1}(c_{j}+\iota _{j})+n-n_{k}+1$
.
Writing
$N_1=\bigcup _{k\ge 0}\{n_k,n_k+1,\ldots ,n_k+c_k\}$
, we have
$$ \begin{align*} \limsup_{n\in N_1, n\rightarrow\infty}\frac{\sharp\{i\leq n,i\in D\}}{n}&\leq\limsup_{k\rightarrow\infty}\bigg(\frac{\sum_{j=0}^{k-1}(c_{j}+\iota_{j})}{n_{k}} +\frac{c_{k}}{n_{k}}\bigg) \\&\le\limsup_{k\rightarrow\infty}\frac{\sum_{j=0}^{k-1}(c_{j}+\iota_{j})}{n_{k}} +\limsup_{k\rightarrow\infty}\frac{c_{k}}{n_{k}} \\&\leq\limsup_{k\rightarrow\infty}\frac{c_{k-1}+\iota_{k-1}}{n_{k}-n_{k-1}}=0, \end{align*} $$
where the third inequality follows by the Stolz–Cesàro theorem and the definition of
$\{c_{k}\},$
and the last equality holds by (3.3) and the definition of
$\iota _{k}.$
-
(ii) If
$n_{k}+lc_{k}\leq n<n_{k}+(l+1)c_{k}$
for some l with
$0< l<\iota _{k},$
then
$$ \begin{align*}\sharp\{i\leq n,i\in D\}=\sum_{j=0}^{k-1}(c_{j}+\iota_{j})+c_{k}+l.\end{align*} $$
-
(iii) If
$n_{k}+\iota _{k}c_{k}\leq n<n_{k+1},$
then
$\sharp \{i\leq n,i\in D\}=\sum _{j=0}^{k}(c_{j}+\iota _{j}).$
By carrying out the same calculations as those following condition (i), we arrive at
$$ \begin{align*}\limsup_{n\in N_1, n\rightarrow\infty}\frac{\sharp\{i\leq n,i\in D\}}{n}=0.\end{align*} $$
This establishes that the density is zero and proves Claim 2.
For
$j\ge 0$
,
$1\leq t\leq \iota _{j}$
, write
$$ \begin{align*}k_{\iota_{0}+\cdots +\iota_{j-1}+t}&=n_{j}+(t-1)c_{j}-\sum_{i=0}^{j-1}(c_{i}+\iota_{i})-1,\\B_{\iota_{0}+\cdots +\iota_{j-1}+t}&= \begin{cases} (\underbrace{11\ldots 11}_{c_{j}+1}) &\mbox{for } t=1,\\ (1) &\mbox{for } 1<t\leq \iota_{j}, \end{cases} \end{align*} $$
and
$$ \begin{align*} B^{\prime}_{\iota_{0}+\cdots +\iota_{j-1}+t}= \begin{cases} (1\underbrace{M\ldots M}_{c_{j}-1}1) &\mbox{for } t=1,\\ (M) &\mbox{for } 1<t\leq \iota_{j}. \end{cases} \end{align*} $$
For any
$p>1,$
$$ \begin{align*}J_{M-1,p,1}(\{k_{n}\},\{B_{n}\})\subset E_{M}(\{n_k\}, \{c_k\}),\quad J_{M,p,M}(\{k_{n}\},\{B^{\prime}_{n}\})\subset E^{\prime}_{M}(\{n_k\}, \{c_k\}).\end{align*} $$
By Claims 1 and 2, and Lemmas 2.2, 2.3 and 2.4,
$$ \begin{align*}\dim_{H}E_{\alpha,\beta}\ge\dim_{H}(E_{M}(\{n_k\}, \{c_k\})\times E^{\prime}_{M}(\{n_k\}, \{c_k\}))\geq\dim_{H}J_{M-1}+\dim_{H}J_{M}.\end{align*} $$
We complete the proof of Case 1 by letting
$M\to \infty $
and using Lemma 2.1.
Similar arguments apply in the remaining cases. Here, we only give the constructions for the proper sequences
$\{n_k\}_{k\geq 0}$
and
$\{c_k\}_{k\geq 0}.$
It is easy to check that the corresponding D is of density zero and
$E_{M}(\{n_k\},\{c_k\})\times E^{\prime }_{M}(\{n_k\},\{c_k\})$
is a subset of
$E_{\alpha ,\beta }$
in each case.
Case 2:
$0<\alpha =\beta <\infty .$
Take
$n_{k+1}=n_{k}+\lfloor (\log n_{k})^{2} \rfloor , c_{k}=\lfloor \alpha \log n_{k}\rfloor \text { for } k\geq 0.$
Case 3:
$\alpha =0, 0<\beta <\infty .$
Take
$n_{k+1}=n^{k}_{k}, c_{k}=\lfloor \beta \log n_{k}\rfloor ~\text {for } k\geq 0.$
Case 4:
$\alpha =0, \beta =\infty .$
Take
$n_{k+1}=n^{n_{k}}_{k}, c_{k}=\lfloor \sqrt {n_{k}\log n_{k}}\rfloor ~\text {for } k\geq 0.$
Case 5:
$0<\alpha <\beta =\infty .$
Take
$n_{k+1}=\lfloor n^{\log n_{k}}_{k}\rfloor , c_{k}=\lfloor \alpha (\log n_{k})^{2}\rfloor ~\text {for } k\geq 0.$
Case 6:
$\alpha =\beta =\infty .$
Take
$n_{k+1}=n^{n_{k}}_{k}, c_{k}=\lfloor (n_{k}\log n_{k})^{2}\rfloor ~\text {for } k\geq 0.$
Case 7:
$\alpha =\beta =0.$
Take
$n_{k+1}=(n_{k})^{2}, c_{k}=\lfloor \sqrt {\log n_{k}}\rfloor ~\text {for } k\geq 0.$
Remark 3.1. If
$\Lambda $
is finite, then
$\dim _{H}J=\dim _{P}J$
(see [Reference Mauldin and Urbański15]). From Section 3.1,
$\dim _{H}E_{\alpha ,\beta }=2\dim _{H}J.$
3.2
$E_{\alpha ,\beta }$
is dense in
$J\times J$
For
$0\leq \alpha \leq \beta \leq \infty ,$
there exists
$(x',y')\in J\times J$
such that
$$ \begin{align*}\liminf_{n\rightarrow\infty}\frac{-\log m_{n}(x',y')}{\log n}=\alpha,\quad \limsup_{n\rightarrow\infty}\frac{-\log m_{n}(x',y')}{\log n}=\beta.\end{align*} $$
For any
$(x,y)\in J\times J,$
we can choose a sequence of points
$\{(\widehat {x}_{k},\widehat {y}_{k})\}_{k\geq 1}$
such that
$$ \begin{align*}\widehat{x}_{k}=[\omega_{1}(x),\ldots,\omega_{k}(x),\omega_{k}(x'),\ldots] \quad \text{and} \quad \widehat{y}_{k}=[\omega_{1}(y),\ldots,\omega_{k}(y),\omega_{k}(y'),\ldots].\end{align*} $$
Clearly,
$(\widehat {x}_{k},\widehat {y}_{k})\rightarrow (x,y)$
as
$k\rightarrow \infty .$
In fact, we can check that for any
$k\geq 1,$
$$ \begin{align*}\liminf_{n\rightarrow\infty}\frac{-\log m_{n}(\widehat{x}_{k},\widehat{y}_{k})}{\log n}=\alpha,\quad \limsup_{n\rightarrow\infty}\frac{-\log m_{n}(\widehat{x}_{k},\widehat{y}_{k})}{\log n}=\beta.\end{align*} $$
Thus,
$E_{\alpha ,\beta }$
is dense in
$J\times J.$
4 Proof of Theorem 1.3
4.1 Estimation of
$\mathbf{dim} _{H}\ F_{\alpha ,\beta }$
When
$0<\alpha <\beta <\infty ,$
we can check that the Cantor subset
$E_{M}(\{n_k\}, \{m_k\})\times E^{\prime }_{M}(\{n_k\}, \{c_k\})$
constructed in Section 3.1 is a subset of
$F_{\alpha ,\beta }.$
Similarly,
$\dim _{H}F_{\alpha ,\beta }\geq 2\dim _{H}J.$
The other cases can be proved by some minor modifications.
4.2
$F_{\alpha ,\beta }$
is dense in
$J\times J$
As we have argued in Section 3.2, we can get the corresponding results for
$F_{\alpha ,\beta }.$
We omit the details.
Acknowledgment
The authors would like to thank the referees for their valuable comments and suggestions that led to the improvement of the manuscript.
