1 Introduction
Let $(M,\rho )$ be a metric space. Suppose that the Hausdorff–Besicovitch dimension $\dim _{H}$ [8] is well defined in $(M,\rho )$. A transformation $f:M\to M$ is called dimension-preserving transformation [13] or $\mathit{DP}$-transformation if
Let $G(M,\dim _{H})$ be the set of all $\mathit{DP}$-transformations defined on $(M,\rho )$. It is easy to see that G forms a group w.r.t. the composition of transformations. It is well known that any bi-Lipschitz transformation belongs to this group [8]. However, G is essentially wider than the group of all bi-Lipschitz transformations. In 2004, some sufficient conditions for belonging of distribution functions of random variable with independent s-adic digits to group G was proved by G. Torbin et al. [2]. There exist a lot of $\mathit{DP}$-functions that are not bi-Lipschitz.
Sufficient conditions for distribution functions of random variables with independent s-adic digits to be $\mathit{DP}$ have been found by G. Torbin [13] in 2007. These conditions were generalized for Q by G. Torbin [14] and later for ${Q}^{\ast }$- and $\tilde{Q}$-expansions by S. Albeverio, V. Koshmanenko, M. Pratsiovytyi, and G. Torbin [3, 4].
Recently, G. Torbin and M. Ibragim proved rather general sufficient conditions for distribution functions of random variables with independent $\tilde{Q}$-digits to be in $\mathit{DP}$-class. The notion of fine covering system faithfulness for $\dim _{H}$ calculation [5] plays an important role in the proof of these conditions. This notion gives us the possibility to consider coverings by sets from some family Φ and to be sure that a “dimension” calculated in such a way is equal to $\dim _{H}$. Faithfulness of the family of all s-adic cylinders (if s is fixed) have been proven by Billingsley [6] in 1961. Faithfulness of the family of Q-cylinders have been proven by M. Pratsiovytyi and A. Turbin [16] in 1992, and faithfulness of the family of ${Q}^{\ast }$-cylinders (under the condition of separation from zero of the corresponding coefficients) have been proven by S. Albeverio and G. Torbin [1] in 2005. It is necessary to remark that the last result can be easily generalized to $\tilde{Q}$-expansion under a similar condition.
In 1982, C. Tricot [15] introduced the notion of packing dimension $\dim _{P}$. This dimension is in some sense dual to the Hausdorff–Besicovitch dimension: the definition of $\dim _{H}$ of a set F is based on ε-coverings of this figure, but the definition of $\dim _{P}$ is based on ε-packings (the countable sets of disjoint open balls $B_{k}(r_{k},c_{k})$, $k\in \mathbb{N}$, with radii $r_{k}\leqslant \varepsilon $ and centers $c_{k}\in F$). The packing dimension has all “good” properties of a fractal dimension, such as the countable stability. Therefore, proving or disproving similar results for $\dim _{P}$ is important. For example, we consider the group of packing-dimension-preserving transformations (or $\mathit{PDP}$-transformations).
There are a lot of problems with proving of many conjectures for $\dim _{P}$ because work with packings is essentially more complicated than work with coverings [10].
These problems are solving bit by bit. For example, M. Das [7] has proven the Billingsley theorem for packing dimension; J. Li [9] obtained some sufficient conditions for distribution functions of random variables with independent $\tilde{Q}$-digits to be in $\mathit{PDP}$-class. Namely, J. Li has proven the following theorem.
Theorem 1.1.
Let $F_{\xi }$ be the distribution function of a random variable ξ with independent $\tilde{Q}$-representation. If $\inf _{i,j}q_{ij}=q_{\ast }>0$ and $\inf _{i,j}p_{ij}=p_{\ast }>0$, then $F_{\xi }$ preserves the packing dimension if and only if
\[ \underset{k\to \infty }{\limsup }\frac{h_{1}+h_{2}+\cdots +h_{k}}{b_{1}+b_{2}+\cdots +b_{k}}=1,\]
where $h_{j}=-{\sum _{i=1}^{n_{j}}}p_{ij}\ln p_{ij}$ and $b_{j}=-{\sum _{i=1}^{n_{j}}}p_{ij}\ln q_{ij}$.
In Remark 4.2 at the and of article [9], we read: “The conditions $\inf _{i,j}q_{ij}=q_{\ast }>0$ and $\inf _{i,j}p_{ij}=p_{\ast }>0$ play an important role in the proof of the theorem. Open question: What can we say about the topic if we remove these conditions?”
S. Albeverio, M. Pratsiovytyi, and G. Torbin [3] removed the condition $\inf _{i,j}p_{ij}=p_{\ast }>0$ in a similar situation for $\mathit{DP}$-transformations.
In case of packing dimension, the approach of [3] is complicated because it requires appropriate results about the fine packing system faithfulness for packing dimension calculation. Even the definition of the fine packing system faithfulness is a problem because centers of all balls in packings should be in the set the dimension of which is calculated.
The aim of this paper is to propose some alternative definition of the packing dimension, uncentered packing dimension or $\dim _{P(\mathit{unc})}$. In the proposed definition, the condition “the centers of balls should be in the figure the dimension of which is calculated” in the definition of $\dim _{P}$ is replaced by “every ball should have a nonempty intersection with the figure.” We prove that, in some wide class of metric spaces (including ${\mathbb{R}}^{n}$), the value of packing dimension with uncentered balls is matching to the value of classical packing dimension. Introduction of the fine packing system faithfulness notion is very simple in the case of proposed definition. It allows us to prove faithfulness (under the condition of separation from zero of the coefficients) of a $\tilde{Q}$-cylinder system and sufficient conditions for the distribution function of a random variable with independent $\tilde{Q}$-digits to be in the $\mathit{PDP}$-class. The corresponding theorem is the main result of the paper.
2 Packing dimension
Let us recall the definition of packing dimension in the form given, for example, in [8].
Definition 2.1.
Let $E\subset M$ and $\varepsilon >0$. A finite or countable family $\{E_{j}\}$ of open balls is called an ε-packing of a set E if
Definition 2.2.
Let $E\subset M$, $\alpha \geqslant 0$, $\varepsilon >0$. Then the α-dimensional packing premeasure of a bounded set E is defined by
where the supremum is taken over all at most countable ε-packings $\{E_{j}\}$ of E (if $E_{j}=\varnothing $ for all j, then ${\mathcal{P}_{\varepsilon }^{\alpha }}(E)=0$).
3 Uncentered packing dimension
Definition 3.1.
Let $E\subset M$ and $\varepsilon >0$. A finite or countable family $\{E_{j}\}$ of open balls is called an uncentered ε-packing of a set E if
Definition 3.2.
Let $E\subset M$, $\alpha \geqslant 0$, $\varepsilon >0$. Then the uncentered α-dimensional packing premeasure of a bounded set E is defined by
where the supremum is taken over all at most countable uncentered ε-packings $\{E_{i}\}$ of E.
Definition 3.4.
Uncentered α-dimensional packing measure is defined by
where the infimum is taken over all at most countable coverings $\{E_{j}\}$ of E, $E_{j}\subset \mathbf{M}$.
Remark 3.2.
If $(M,\rho )={\mathbb{R}}^{1}$ and $\alpha =1$, then the α-dimensional packing measure and uncentered α-dimensional packing measure are the Lebesgue measure.
Theorem 3.1.
Let $(M,\rho )$ be a metric space. Let $C\in \mathbb{N}$. If for all $r>0$ and for any open ball I with $|I|=8r$, there exist at most $N(I)$ balls $I_{i}$, $i\in \{1,\dots ,\hspace{2.5pt}N(I)\}$ such that $I_{i}\subset I$, $i\in \{1,\dots ,N(I)$, $|I_{i}|=r$, $i\in \{1,\dots ,N(I)\}$, and $N(I)\le C$. Then
Proof.
Step 1. Let us prove the inequality $\dim _{P(\mathit{unc})}(E)\geqslant \dim _{P}(E)$.
By the definitions and supremum property we have
By the limit property of inequalities we have
Hence,
Let $\dim _{P(\mathit{unc})}(E)=\alpha _{0}$. By the definition of $\dim _{P(\mathit{unc})}(E)$ we have
\[ \forall \varepsilon >0,\hspace{1em}{\mathcal{P}_{(\mathit{unc})}^{\alpha _{0}+\varepsilon }}(E)=0.\]
Therefore,
and, consequently,
Hence, it follows that $\dim _{P(\mathit{unc})}(E)\geqslant \dim _{P}(E)$, which is our claim.
Step 2. Let us show that $\dim _{P(\mathit{unc})}(E)\leqslant \dim _{P}(E)$.
If $\dim _{P(\mathit{unc})}(E)=0$, then the statement is true.
Let us consider the case $\dim _{P(\mathit{unc})}(E)\ne 0$. Fix $0<t<s<\dim _{P(\mathit{unc})}(E)$.
Since $s<\dim _{P(\mathit{unc})}(E),$ we have
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle {\mathcal{P}_{(\mathit{unc})}^{s}}(E)=+\infty ,\\{} & \displaystyle {\mathcal{P}_{0(\mathit{unc})}^{s}}(E)=+\infty .\end{array}\]
Therefore,
From this and from the supremum property, it follows that there is an uncentered packing $V:=\{E_{i}\}$ of the set E with
Let us divide the packing V into classes
Let $n_{k}$ be the number of balls $V_{k}$. We will show that
To obtain a contradiction, suppose that
Then
\[ \sum \limits_{i}|E_{i}{|}^{s}<\sum \limits_{k}{2}^{-ks}\cdot n_{k}<\sum \limits_{k}{2}^{-ks}\cdot {2}^{kt}\big(1-{2}^{t-s}\big)=\big(1-{2}^{t-s}\big)\cdot \sum \limits_{k}{\big({2}^{t-s}\big)}^{k}=1,\]
which contradicts our assumption (1).Therefore, such $k_{0}$ exists. Let us consider $V_{k_{0}}$. We denote by $A_{1},A_{2},\dots ,A_{n_{k_{0}}}$ the balls in $V_{k_{0}}$, that is,
Fix $r:={2}^{-k_{0}-1}$. Then the radius of any $A_{i}$ is less than r. Let $T_{i}$ be a point of $A_{i}$ such that $T_{i}\in A_{i}\cap E$. Let ${V^{\prime }}$ be the set of balls with the centers $T_{i}$ and radius r, that is,
Fix
Let us divide the set ${V^{\prime }}$ into classes $K_{1},K_{2},\dots ,K_{l}$ as follows.
-
1. Let us take a ball ${A^{\prime }_{j_{1}}}={A^{\prime }_{1}}$ and put it in $K_{1}$ together with all other balls ${A^{\prime }_{i}}\in {V^{\prime }}$ such that ${A^{\prime }_{i}}\cap {A^{\prime }_{j_{1}}}\ne \varnothing $.
-
2. Let us take an arbitrary ball ${A^{\prime }_{j_{2}}}\in {V^{\prime }}\setminus K_{1}$ and put it in $K_{2}$ together with all other balls ${A^{\prime }_{i}}\in {V^{\prime }}\setminus K_{1}$ such that ${A^{\prime }_{i}}\cap {A^{\prime }_{j_{2}}}\ne \varnothing $.
-
3. Let us continue this way until ${V^{\prime }}\setminus (K_{1}\cup K_{2}\cup \cdots \cup K_{l})\ne \varnothing $. Since the number of elements in a set ${V^{\prime }}$ is a finite, we can find such a number l.
Now suppose that the balls ${A^{\prime }_{i}}$ and ${A^{\prime }_{j}}$ intersect each other. In other words, $\rho (T_{i},T_{j})\leqslant 2r$. Therefore, $A_{j}\subset {A_{i}^{\ast }}$.
The radius of $A_{j}$ is greater than $r/2$. By the theorem condition, there are no more than C disjoint balls with radius $r/2$ in a ball with radius $4r$
Therefore, there are no more than C balls in any class $K_{i}$.
Moreover, in the case $i<m$, the balls ${A^{\prime }_{j_{i}}}$ and ${A^{\prime }_{j_{m}}}$ do not intersect each other. Indeed, suppose otherwise. Then ${A^{\prime }_{j_{m}}}$ is in a class $K_{i}$ or in a class with number less than i.
Hence,
\[ {V^{\prime\prime }}=\big\{{A^{\prime }_{j_{1}}},{A^{\prime }_{j_{2}}},\dots ,{A^{\prime }_{j_{l}}}\big\}\]
is a centered packing of a set E, and the t-volume of this packing is less than the t-volume of the uncentered packing $V_{k_{0}}$ no more than C times. Therefore,
From this it follows that
By the inequality ${2}^{-k_{0}}<r$ we get
Consequently, as $r\to 0$, we get the inequality
Let us show that ${\mathcal{P}}^{t}(E)\geqslant \frac{1-{2}^{t-s}}{C}$. Recall the definition
\[ {\mathcal{P}}^{t}(E)=\inf \bigg\{\sum \limits_{j}{\mathcal{P}_{0}^{t}}(E_{j}):E\subset \bigcup E_{j}\bigg\},\]
where the infimum is taken over all at most countable coverings $E_{j}$ of a set E.Let $\{E_{j}\}$ be an at most countable covering of E. Since $\dim _{P(\mathit{unc})}(E)>s$, there is $j_{0}$ such that $\dim _{P(\mathit{unc})}(E_{j_{0}})>s$ (by the countable stability of the packing dimension $\dim _{P(\mathit{unc})}$). In other words, we have
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle {\mathcal{P}_{(\mathit{unc})}^{s}}(E_{j_{0}})=+\infty ,\\{} & \displaystyle {\mathcal{P}_{0(\mathit{unc})}^{s}}(E_{j_{0}})=+\infty .\end{array}\]
We conclude by the part of the theorem already proved for E that
and
But the previous inequality is true for an arbitrary covering $\{E_{j}\}$ of a set E and for the infimum for all coverings. Therefore,
and
Since $t\text{-}\dim _{P(\mathit{unc})}(E)$ can be approximated by 0, we get $\dim _{P}(E)\geqslant \dim _{P(\mathit{unc})}(E)$, which completes the proof. □Proof.
Let $B_{8r}$ be a ball with radius $8r$, $B_{r}$ be a ball with radius r, and λ be the n-dimensional Lebesgue measure. Then
Therefore, we can put no more than $C={8}^{n}$ disjoint balls with radii r in a ball with radius $8r$, which completes the proof. □
3.1 Packing dimension with respect to the family of sets
Let Φ be a family of balls in a metric space $(M,\rho )$.
Definition 3.6.
Let $E\subset M$, $\alpha \geqslant 0$, $\varepsilon >0$. Then the α-dimensional packing premeasure of a bounded set E with respect to Φ is defined by
where the supremum is taken over all uncentered ε-packings $\{E_{i}\}\subset \varPhi $ of E (if $\{E_{i}\}=\varnothing $, then ${\mathcal{P}_{\varepsilon }^{\alpha }}(E,\varPhi )=0$).
Definition 3.8.
The α-dimensional packing measure w.r.t. Φ is defined by
where the infimum is taken over all at most countable coverings $\{E_{j}\}$ of E, $E_{j}\subset \mathbf{M}$.
Remark 3.3.
In the definition of $\dim _{P}(E,\varPhi )$, we used uncentered packing. But we will denote this dimension without index $(\mathit{unc})$ because:
4 Faithfulness of the open balls families for packing dimension calculation
Definition 4.1.
Suppose that some open balls family Φ satisfies the following condition: for all $E\subset M$, $\dim _{P(\mathit{unc})}(E,\varPhi )=\dim _{P(\mathit{unc})}(E).$ Then Φ is said to be faithful for uncentered packing dimension calculation.
Remark 4.1.
The notion of faithfulness is introduced for the Hausdorff–Besicovitch dimension $\dim _{H}$ [11]. It is clear that
Theorem 4.1 (The sufficient condition for the open-ball family to be faithful for packing dimension calculation).
Suppose that
Then Φ is a faithful open-ball family for packing dimension calculation.
Proof.
Let E be any set, $\alpha \geqslant 0$, and $r>0$. Let $\{E_{i}\}=\{(a_{i};b_{i})\}$ be a family of disjoint intervals such that $\frac{a_{i}+b_{i}}{2}\in E$ and $b_{i}-a_{i}<r$.
Then the following inequality holds:
Taking the supremum (over all sets of intervals $\{E_{i}\}$ satisfying the previous conditions), we have
Any set of intervals $\{\varDelta (a_{i},b_{i})\}$ satisfies the conditions from the ${\mathcal{P}_{r(\mathit{unc})}^{\alpha }}(E,\varPhi )$ definition. So,
\[ \underset{\{E_{i}\}}{\sup }{\big|\varDelta (a_{i},b_{i})\big|}^{\alpha }\cdot {C}^{\alpha }\leqslant {\mathcal{P}_{r(\mathit{unc})}^{\alpha }}(E,\varPhi )\cdot {C}^{\alpha }.\]
Therefore,
Taking the limit of both sides, we have
Taking the infimum over all possible coverings of the set E, we have
\[ {\mathcal{P}}^{\alpha }(E)\leqslant {\mathcal{P}_{(\mathit{unc})}^{\alpha }}(E,\varPhi )\cdot {C}^{\alpha }\]
and
Since $[0;1]\subset {\mathbb{R}}^{1}$, it follows that
and
Using
we obtain that Φ is a faithful open-ball family for the packing dimension calculation. □5 Sufficient conditions for $\tilde{Q}$-expansion cylindric interval family to be faithful
The $\tilde{Q}$-expansion of real numbers is a generalization of s-expansion and Q-expansion and was described, for example, in [4].
Proof.
Let $\tilde{Q}_{0}$ be the set of $\tilde{Q}$-rational points, and ${E^{\prime }}$ be any subset of $[0;1]$. Let $E={E^{\prime }}\setminus \tilde{Q}_{0}$. Since $\tilde{Q}_{0}$ is countable, it follows that
\[ \dim _{P(\mathit{unc})}(\tilde{Q}_{0})=0,\hspace{2em}\dim _{P(\mathit{unc})}(\tilde{Q}_{0},\varPhi )=0,\]
and
\[ \dim _{P(\mathit{unc})}\big({E^{\prime }}\big)=\dim _{P(\mathit{unc})}(E),\hspace{2em}\dim _{P(\mathit{unc})}\big({E^{\prime }},\varPhi \big)=\dim _{P(\mathit{unc})}(E,\varPhi ).\]
The proof is completed by showing that
for every set $E\subset [0;1]$ if E does not contain $\tilde{Q}$-rational points.Let $(a;b)\subset [0;1]$. Let $\varDelta (a,b)$ be the $\tilde{Q}$-cylindric interval of the minimal rank such that
Denote the rank of $\varDelta (a,b)$ by k. Since this rank is minimal, it follows that $(a;b)$ is a subset of one or two cylinders with rank $k-1$. Let us denote the cylinder with rank $k-1$ that contains $\varDelta (a,b)$ by ${\varDelta ^{\prime }}$. If the second cylinder exists, then we denote it by ${\varDelta ^{\prime\prime }}$.
Summary of the two cases. For every interval $(a;b)$, there exists a $\tilde{Q}$-cylindric interval $\varDelta (a,b)$ such that $\frac{a+b}{2}\in \varDelta (a;b)$ and
It follows that the family Φ satisfies the conditions of Theorem 4.1 and is faithful for packing dimension calculation. □
Corollary 5.1.
Let Φ be a family of ${Q}^{\ast }$-cylinders under the condition $\inf _{i,j}q_{ij}>0$. Then Φ is faithful for packing dimension calculation.
6 Proof of the main result
To prove the main result, we need the following two lemmas.
Lemma 6.1.
Let $\tilde{Q}$ be the matrix $\| q_{ik}\| $, $i\in \mathbb{N}$, $k\in \{0,1,\dots ,N_{k}-1\}$. If
\[ \underset{i\to \infty }{\lim }\frac{\ln q_{i_{k}k}}{\ln (q_{i_{1}1}q_{i_{2}2}\dots q_{i_{k-1}(k-1)})}=0\]
for every sequence $(i_{k})$, then the open-ball family Φ of the respective expansion cylinder interiors is faithful for packing dimension calculation.
Proof.
Let us fix a set $E\subset [0;1]$. Let us fix any numbers $m\in \mathbb{N}$, $\delta >0$ and consider the following sets:
Fix some value m and consider any set $W_{m,\delta }$ corresponding to this value. There exists $\varepsilon >0$ such that $|c_{m}|\geqslant \varepsilon $ for any cylinder $c_{m}$ of rank m. Consider the centered ε-packing of the set $W_{m,\delta }$ by intervals $E_{j}$.
For every interval $E_{j}$, there exists a cylindric interval $\varDelta (E_{j})$ such that:
We will say that the cylinder ${\varDelta ^{\prime }}(E_{j})$ is the “father” of $\varDelta (E_{j})$ if ${\varDelta ^{\prime }}(E_{j})\supset \varDelta (E_{j})$ and the rank of ${\varDelta ^{\prime }}(E_{j})$ is equal to $i_{j}-1$. It is obvious that $|{\varDelta ^{\prime }}(E_{j})|\geqslant \frac{|E_{j}|}{2}$. Therefore,
Let us estimate the α-volume of packing of the set E by intervals $E_{j}$:
This inequality is equivalent to
\[ \sum \limits_{k}|E_{j}{|}^{\alpha }\leqslant \sum \limits_{k}{\big|\varDelta (E_{j})\big|}^{\alpha -\delta }\cdot {\big|\varDelta (E_{j})\big|}^{\delta }\cdot {\bigg(\frac{2}{q_{i_{k_{j}}k_{j}}(x_{j})}\bigg)}^{\alpha }.\]
Let us estimate the expression
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \ln \bigg({\big|\varDelta (E_{j})\big|}^{\delta }\cdot {\bigg(\frac{2}{q_{i_{k_{j}}k_{j}}(x_{j})}\bigg)}^{\alpha }\bigg)\\{} & \displaystyle \hspace{1em}=\delta \ln \big(q_{i_{1}1}(x)q_{i_{2}2}(x)\dots q_{i_{k_{j}-1}(k_{j}-1)}(x)\big)+\alpha \ln 2-\alpha \ln q_{i_{k_{j}}k_{j}}(x_{j}).\end{array}\]
Since $x_{j}\in W_{m,\delta }$, it follows that
\[ \delta \ln \big(q_{i_{1}1}(x)q_{i_{2}2}(x)\dots q_{i_{k_{j}-1}(k_{j}-1)}\big)\leqslant \ln q_{i_{k_{j}}k_{j}}(x_{j}).\]
Therefore,
\[ \ln \bigg({\big|\varDelta (E_{j})\big|}^{\delta }\cdot {\bigg(\frac{2}{q_{i_{k_{j}}k_{j}}(x_{j})}\bigg)}^{\alpha }\bigg)\leqslant \alpha \ln 2+(1-\alpha )\ln q_{i_{k_{j}}k_{j}}(x_{j})\leqslant \alpha \ln 2.\]
Thus, we have
\[ \sum \limits_{j}|E_{j}{|}^{\alpha }\leqslant 2\sum \limits_{j}{\big|\varDelta (E_{j})\big|}^{\alpha -\delta }.\]
Take the suprema over all possible centered packings $\{E_{j}\}$ of both parts of the previous inequality:
\[ {\mathcal{P}_{\varepsilon }^{\alpha }}(W_{m,\delta })\leqslant 2{\mathcal{P}_{\varepsilon (\mathit{unc})}^{\alpha -\delta }}(W_{m,\delta },\varPhi ).\]
Take the limit as $\varepsilon \to 0$:
\[ {\mathcal{P}_{0}^{\alpha }}(W_{m,\delta })\leqslant 2{\mathcal{P}_{0(\mathit{unc})}^{\alpha -\delta }}(W_{m,\delta },\varPhi ).\]
We obtain that
\[ {\mathcal{P}}^{\alpha }(W_{m,\delta })\leqslant 2{\mathcal{P}_{(\mathit{unc})}^{\alpha -\delta }}(W_{m,\delta },\varPhi ).\]
Denote $\alpha _{0}=\dim _{P}(W_{m,\delta })$. Then for all $\alpha <\alpha _{0}$, the left part is equal to infinity. Thus, for all $\alpha <\alpha _{0}$, the right part is equal to infinity too. It follows that
and
Using the definition of $W_{m,\delta }$, we get
Now, by packing dimension countable stability,
Since δ can be arbitrarily small,
To complete the proof, it remains to note that E is any subset of $[0;1]$. Thus, Φ is faithful. □Lemma 6.2.
Let Φ be a family of $\tilde{Q}$-expansion cylinders under the condition $\inf q_{ij}>0$. Let $F_{\xi }$ be a distribution function of a random variable ξ with independent $\tilde{Q}$-digits. Assume that the following condition holds:
where $\varDelta _{n}(x)$ is the n-rank cylinder that contains x.
(2)
\[ \underset{n\to \infty }{\lim }\frac{\ln \lambda (F(\varDelta _{n}(x)))}{\ln \lambda (\varDelta _{n}(x))}=1,\hspace{1em}\forall x\in [0;1],\]
Then ${\varPhi ^{\prime }}=F(\varPhi )$ is faithful for packing dimension calculation.
Proof.
${\varPhi ^{\prime }}$ is the family of cylinders for some $\tilde{Q}$-expansion. Denote this expansion by ${\tilde{Q}^{\prime }}$ and the corresponding numbers $q_{ij}$ by ${q^{\prime }_{ij}}$.
It is not clear that condition $\inf {q^{\prime }_{ij}}>0$ holds, so we cannot Theorem 5.1.
Let us show that the conditions of Lemma 6.1 hold for this expansion. We have
\[ F\big({\varDelta _{a_{1}a_{2}\dots a_{n}}^{\tilde{Q}}}(x)\big)={\varDelta _{a_{1}a_{2}\dots a_{n}}^{{\tilde{Q}^{\prime }}}}(x)\]
and
\[ \ln \lambda \big(F\big(\varDelta _{n}(x)\big)\big)=\ln \big({q^{\prime }_{1a_{1}}}{q^{\prime }_{2a_{2}}}\dots {q^{\prime }_{na_{n}}}\big).\]
Denote
\[ M=\underset{i\to \infty }{\limsup }\frac{\ln {q^{\prime }_{ij_{i}}}}{\ln ({q^{\prime }_{1j_{1}}}{q^{\prime }_{2j_{2}}}\dots {q^{\prime }_{(i-1)j_{i-1}}})}.\]
To estimate M, we need the following equation:
\[ \underset{n\to 0}{\lim }\frac{\ln \lambda (F(\varDelta _{n}(x)))}{\ln \lambda (\varDelta _{n}(x))}=\underset{n\to 0}{\lim }\frac{\ln ({q^{\prime }_{1j_{1}}}{q^{\prime }_{2j_{2}}}\dots {q^{\prime }_{(i-1)j_{i-1}}})+\ln {q^{\prime }_{ij_{i}}}}{\ln (q_{1j_{1}}q_{2j_{2}}\dots q_{(i-1)j_{i-1}})+\ln q_{ij_{i}}}.\]
Dividing the nominator and denominator of the last fraction by $\ln (q_{1j_{1}}q_{2j_{2}}\dots q_{(i-1)j_{i-1}})$, we obtain
\[ \underset{n\to 0}{\lim }\frac{1+\frac{\ln {q^{\prime }_{ij_{i}}}}{\ln ({q^{\prime }_{1j_{1}}}{q^{\prime }_{2j_{2}}}\dots {q^{\prime }_{(i-1)j_{i-1}}})}}{\frac{\ln (q_{1j_{1}}q_{2j_{2}}\dots q_{(i-1)j_{i-1}})}{\ln ({q^{\prime }_{1j_{1}}}{q^{\prime }_{2j_{2}}}\dots {q^{\prime }_{(i-1)j_{i-1}}})}+\frac{\ln q_{ij_{i}}}{\ln ({q^{\prime }_{1j_{1}}}{q^{\prime }_{2j_{2}}}\dots {q^{\prime }_{(i-1)j_{i-1}}})}}=\frac{1+M}{1+0}=1\hspace{1em}\Rightarrow \hspace{1em}M=0.\]
It follows that ${\tilde{Q}^{\prime }}$ satisfies the conditions of Lemma 6.1, and therefore ${\varPhi ^{\prime }}$ is faithful. □Proof of the main result.
Let us show that if $F_{\xi }$ is $\mathit{PDP}$, then $\dim _{P}(\mu _{\xi })=1$.
Assume the converse. Then there exists a set $E_{\alpha }$ such that $\mu _{\xi }(E_{\alpha })=1$ and $\dim _{P}(E_{\alpha })=\alpha $. Consider $F_{\xi }(E_{\alpha })$. Since $\mu _{\xi }(E_{\alpha })=1$, we have $\lambda (E_{\alpha })=1$, and thus $\dim _{P}(F_{\xi }(E_{\alpha }))=1$.
We obtain the following inequality:
and this contradicts the assumption that $F_{\xi }$ is $\mathit{PDP}$. Therefore, we will show that if $F_{\xi }$ is $\mathit{PDP}$, then $\dim _{P}(\mu _{\xi })=1$.
The next part of the proof consists of two steps:
Let ε be some positive number such that $\varepsilon <\frac{1}{2}q_{\min }$. Consider the following sets:
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle {T_{\varepsilon ,k}^{+}}=\big\{j:j\in \mathbb{N},j\leqslant k,|p_{ij}-q_{ij}|\leqslant \varepsilon ,\hspace{2.5pt}i\in \{0,1,\dots ,s-1\}\big\},\\{} & \displaystyle {T_{\varepsilon ,k}^{-}}=\{1,2,\dots ,k\}\setminus {T_{\varepsilon ,k}^{+}},\\{} & \displaystyle T=\bigg\{k:k\in \mathbb{N},p_{k}<\frac{1}{2}q_{\min }\bigg\},\\{} & \displaystyle T_{k}=T\cap \{1,2,\dots ,k\},\\{} & \displaystyle T_{\varepsilon ,k}={T_{\varepsilon ,k}^{-}}\setminus T_{k}.\end{array}\]
Step 1. Let us show that if $\dim _{P}(\mu _{\xi })=1$ and $B=0$, then $F_{\xi }$ is $\mathit{PDP}$. Since $B=0$, we see that
Consider the fraction
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \frac{\ln \mu _{\xi }(\varDelta _{a_{1}a_{2}\dots a_{k}(x)})}{\ln \lambda (\varDelta _{a_{1}a_{2}\dots a_{k}(x)})}\\{} & \displaystyle \hspace{1em}=\frac{\sum _{j\in {T_{\varepsilon ,k}^{+}}}\ln p_{a_{j}(x)j}+\sum _{j\in T_{\varepsilon ,k}}\ln p_{a_{j}(x)j}+\sum _{j\in T_{k}}\ln p_{a_{j}(x)j}}{\sum _{j}\ln q_{a_{j}(x)j}}.\end{array}\]
Split this fraction into three terms. Consider the first term
It is easy to prove that
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \sum \limits_{j\in {T_{\varepsilon ,k}^{+}}}\ln p_{a_{j}(x)j}& \displaystyle \geqslant \sum \limits_{j\in {T_{\varepsilon ,k}^{+}}}\ln (q_{a_{j}(x)j}-\varepsilon )\\{} & \displaystyle =\sum \limits_{j\in {T_{\varepsilon ,k}^{+}}}\bigg(\ln q_{a_{j}(x)j}+\ln \bigg(\frac{q_{a_{j}(x)j}-\varepsilon }{q_{a_{j}(x)j}}\bigg)\bigg)\\{} & \displaystyle \geqslant \sum \limits_{j\in {T_{\varepsilon ,k}^{+}}}(\ln q_{a_{j}(x)j})+|{T_{\varepsilon ,k}^{+}}|\cdot \frac{2\varepsilon }{q_{\min }},\end{array}\]
where $|{T_{\varepsilon ,k}^{+}}|$ is the number of elements in ${T_{\varepsilon ,k}^{+}}$. On the other hand,
\[ \sum \limits_{j\in {T_{\varepsilon ,k}^{+}}}\ln p_{a_{j}(x)j}\leqslant \sum \limits_{j\in {T_{\varepsilon ,k}^{+}}}(\ln q_{a_{j}(x)j})-|{T_{\varepsilon ,k}^{+}}|\cdot \frac{2\varepsilon }{q_{\min }}.\]
Also,
\[ 1+\frac{|{T_{\varepsilon ,k}^{+}}|\cdot 2\varepsilon }{q_{\min }\cdot {\textstyle\sum _{j=0}^{k}}\ln q_{a_{j}(x)j}}\leqslant \underset{k\to \infty }{\lim }\frac{\sum _{j\in {T_{\varepsilon ,k}^{+}}}\ln p_{a_{j}(x)j}}{{\textstyle\sum _{j=0}^{k}}\ln q_{a_{j}(x)j}}\leqslant 1-\frac{|{T_{\varepsilon ,k}^{+}}|\cdot 2\varepsilon }{q_{\min }\cdot {\textstyle\sum _{j=0}^{k}}\ln q_{a_{j}(x)j}}\]
(note that $\frac{|{T_{\varepsilon ,k}^{+}}|\cdot 2\varepsilon }{q_{\min }\cdot {\sum _{j=0}^{k}}\ln q_{a_{j}(x)j}}<0$).Since $q_{\min }\leqslant q_{ij}\leqslant q_{\max }$ and $|{T_{\varepsilon ,k}^{+}}|\leqslant k$, we have
\[ 1+\frac{k\cdot 2\varepsilon }{q_{\min }\cdot k\ln q_{\max }}\leqslant \underset{k\to \infty }{\lim }\frac{\sum _{j\in {T_{\varepsilon ,k}^{+}}}\ln p_{a_{j}(x)j}}{{\textstyle\sum _{j=0}^{k}}\ln q_{a_{j}(x)j}}\leqslant 1-\frac{k\cdot 2\varepsilon }{q_{\min }\cdot k\ln q_{\max }}\]
and
Since ε can be arbitrarily small, it follows that
Similarly,
Therefore,
\[ \frac{\sum _{j\in T_{\varepsilon ,k}}\ln p_{a_{j}(x)j}}{k\ln q_{\min }}\leqslant \frac{|T_{\varepsilon ,k}|\ln (\frac{q_{\min }}{2})}{k\ln q_{\min }}\leqslant \frac{|T_{\varepsilon ,k}|(\ln (q_{\min })+\ln (1/2))}{k\ln q_{\min }},\]
and the second term tends to zero as $k\to \infty $.Consider the third term
It can be estimated by
and this value tends to zero as $k\to \infty $ because $B=0$.
We obtain that
Denote by Φ the cylinder family of given $\tilde{Q}$-expansion. Denote the image of Φ by ${\varPhi ^{\prime }}=F_{\xi }(\varPhi )$.
Using the Billingsley theorem for packing dimension [12], we have
To prove that $\dim _{P}(E)=\dim _{P}(F_{\xi }(E))$, it suffices to prove that Φ and ${\varPhi ^{\prime }}$ are faithful.
Faithfulness of Φ is already proved. Faithfulness of ${\varPhi ^{\prime }}$ was proved in Lemma 1 and Lemma 2. So, we have that $\dim _{P}(E)=\dim _{P}(F_{\xi }(E))$ and $F_{\xi }$ is a $\mathit{PDP}$-transformation.
Step 2. Let us show that if $\dim _{P}(\mu _{\xi })=1$ and $B>0$, then $F_{\xi }$ is not $\mathit{PDP}$.
Similarly to step 1, consider the fraction
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \frac{\mu (\varDelta _{a_{1}a_{2}\dots a_{k}(x)})}{\lambda (\varDelta _{a_{1}a_{2}\dots a_{k}(x)})}\\{} & \displaystyle \hspace{1em}=\frac{\sum _{j\in {T_{\varepsilon ,k}^{+}}}\ln p_{a_{j}(x)j}+\sum _{j\in T_{\varepsilon ,k}}\ln p_{a_{j}(x)j}+\sum _{j\in T_{k}}\ln p_{a_{j}(x)j}}{\sum _{j}\ln q_{a_{j}(x)j}}\end{array}\]
and split it into three terms. It is easy to see that the first term tends to 1 and the second term tends to 0 (as $k\to \infty $). Consider the third term.Since $B>0$, there exists a subsequence $(k_{m})$ such that
Consider the set
\[ L=\left\{x:x=\varDelta _{a_{1}a_{2}\dots a_{k}\dots };\Bigg\{\begin{array}{r@{\hskip0pt}l}& \displaystyle a_{k}\in \{0,1,\dots ,s-1\}\hspace{2.5pt}\text{if}\hspace{2.5pt}k\notin T\\{} & \displaystyle a_{k}=n_{k},\hspace{2.5pt}\text{if}\hspace{2.5pt}k\in T,\hspace{2.5pt}\text{where}\hspace{2.5pt}p_{n_{k}k}=\underset{i}{\min }p_{ik}\end{array}\right\}.\]
Since the digits are in infinitely many places, it follows that $\lambda (L)=0$. But combining
and the formula for $\dim _{P}(\mu _{\xi })$, we have $\dim _{P}(L)=1$. It follows that
\[ \forall x\in L\hspace{1em}\underset{m\to \infty }{\lim }\frac{\ln \mu (\varDelta _{a_{1}a_{2}\dots a_{k_{m}}(x)})}{\ln \lambda (\varDelta _{a_{1}a_{2}\dots a_{k_{m}}(x)})}=1+B.\]
Thus, for any $\delta >0$, there exists $m(\delta )$ such that for all $m>m(\delta )$, we have
\[ 1+B-\delta \leqslant \frac{\ln \mu (\varDelta _{a_{1}a_{2}\dots a_{k_{m}}(x)})}{\ln \lambda (\varDelta _{a_{1}a_{2}\dots a_{k_{m}}(x)})}\leqslant 1+B+\delta .\]
Thus, we have
\[ \underset{k\to \infty }{\liminf }\frac{\ln \mu (\varDelta _{a_{1}a_{2}\dots a_{k}(x)})}{\ln \lambda (\varDelta _{a_{1}a_{2}\dots a_{k}(x)})}\geqslant 1+B-\delta ,\]
and (using the Billingsley theorem for $\dim _{P}$)
that is,
Since the last inequality holds for any δ, it follows that
and $F_{\xi }$ is not a $\mathit{PDP}$-transformation. □
