On packing dimension preservation by distribution functions of random variables with independent $\tilde{Q}$-digits

The article is devoted to finding conditions for the packing dimension preservation by distribution functions of random variables with independent $\tilde{Q}$-digits. The notion of"faithfulness of fine packing systems for packing dimension calculation"is introduced, and connections between this notion and packing dimension preservation are found.


Introduction
Let (M, ρ) be a metric space. Suppose that the Hausdorff-Besicovitch dimension dim H [8] is well defined in (M, ρ). A transformation f : M → M is called dimension-preserving transformation [13] or DP-transformation if Let G(M, dim H ) be the set of all DP-transformations defined on (M, ρ). 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 DP-functions that are not bi-Lipschitz.
Sufficient conditions for distribution functions of random variables with independent s-adic digits to be DP have been found by G. Torbin [13] in 2007. These conditions were generalized for Q by G. Torbin [14] and later for Q * -andQ-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 independentQ-digits to be in DPclass. 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 * -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 toQ-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 ∈ N, with radii r k ε and centers c k ∈ 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 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 independentQ-digits to be in PDP -class. Namely, J. Li has proven the following theorem. Theorem 1.1. Let F ξ be the distribution function of a random variable ξ with inde-pendentQ-representation. If inf i,j q ij = q * > 0 and inf i,j p ij = p * > 0, then F ξ preserves the packing dimension if and only if In Remark 4.2 at the and of article [9], we read: "The conditions inf i,j q ij = q * > 0 and inf i,j p ij = p * > 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 * > 0 in a similar situation for 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 (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 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 aQ-cylinder system and sufficient conditions for the distribution function of a random variable with independentQ-digits to be in the PDP -class. The corresponding theorem is the main result of the paper. Let F ξ be the distribution function of a random variable ξ with independentQrepresentation. Then F ξ preserves the packing dimension if and only if dim P µ ξ = 1; B = 0.

Packing dimension
Let us recall the definition of packing dimension in the form given, for example, in [8].
Remark 2.1. The empty set of balls is a packing of any set.
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 = ∅ for all j, then P α ε (E) = 0). Definition 2.3. The α-dimensional packing quasi-measure of a set E is defined by Definition 2.4. The α-dimensional packing measure is defined by where the infimum is taken over all at most countable coverings {E j } of E, E j ⊂ M.
Remark 3.1. The empty set of balls is an uncentered packing of any set.
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 3.3.
The uncentered α-dimensional packing quasi-measure of a set E is defined by where the infimum is taken over all at most countable coverings is called the uncentered packing dimension of a set E ⊂ M . Proof.
Step 1. Let us prove the inequality dim P (unc) (E) dim P (E). By the definitions and supremum property we have P α r(unc) (E) P α r (E).
By the limit property of inequalities we have and, consequently, Hence, it follows that dim P (unc) (E) dim P (E), which is our claim.
Step 2. Let us show that dim P (unc) (E) dim P (E).
If dim P (unc) (E) = 0, then the statement is true.
Let us consider the case dim P (unc Therefore, ∀r > 0, P s r(unc) (E) = +∞. From this and from the supremum property, it follows that there is an uncentered 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 which contradicts our assumption (1). Therefore, such k 0 exists. Let us consider V k0 . We denote by A 1 , A 2 , . . . , A n k 0 the balls in V k0 , that is, Let V ′ be the set of balls with the centers T i and radius r, that is, . Let us divide the set V ′ into classes K 1 , K 2 , . . . , K l as follows.

Let us take an arbitrary ball
3. Let us continue this way until V ′ \(K 1 ∪K 2 ∪· · ·∪K l ) = ∅. Since the number of elements in a set V ′ is a finite, we can find such a number l.
Now suppose that the balls A ′ i and A ′ j intersect each other. In other words, ρ(T i , T j ) 2r. Therefore, A j ⊂ A * i . 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 ′ ji and A ′ jm do not intersect each other. Indeed, suppose otherwise. Then A ′ jm is in a class K i or in a class with number less than i. Hence, 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 k0 no more than C times. Therefore, From this it follows that By the inequality 2 −k0 < r we get Consequently, as r → 0, we get the inequality 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 (unc) (E) > s, there is j 0 such that dim P (unc) (E j0 ) > s (by the countable stability of the packing dimension dim P (unc) ). In other words, we have P s (unc) (E j0 ) = +∞, P s 0(unc) (E j0 ) = +∞.
We conclude by the part of the theorem already proved for E that But the previous inequality is true for an arbitrary covering {E j } of a set E and for the infimum for all coverings. Therefore, Since t-dim P (unc) (E) can be approximated by 0, we get dim P (E) dim P (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.

Packing dimension with respect to the family of sets
Let Φ be a family of balls in a metric space (M, ρ). Definition 3.6. Let E ⊂ M , α 0, ε > 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 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 ⊂ M.

Remark 3.3.
In the definition of dim P (E, Φ), we used uncentered packing. But we will denote this dimension without index (unc) because: 1. We will work in R n . In this space, centered and uncentered packing dimensions are equal; 2. The centered packing dimension w.r.t. some family of balls is not defined.

Proof. Let Φ 0 be the family of all open balls of M . Then
Since Φ ⊆ Φ 0 , by the supremum property we have

By the inequality for packing premeasures it follows that
which proves the theorem.  [11]. It is clear that

Then Φ is a faithful open-ball family for packing dimension calculation.
Proof. Let E be any set, α 0, and r > 0. Let {E i } = {(a i ; b i )} be a family of disjoint intervals such that ai+bi 2 ∈ 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 Therefore, P α r (E) P α r(unc) (E, Φ) · C α . Taking the limit of both sides, we have Taking the infimum over all possible coverings of the set E, we have Since [0; 1] ⊂ R 1 , it follows that dim P (E) = dim P (unc) (E) and dim P (unc) (E) dim P (unc) (E, Φ).
we obtain that Φ is a faithful open-ball family for the packing dimension calculation.

Sufficient conditions forQ-expansion cylindric interval family to be faithful
TheQ-expansion of real numbers is a generalization of s-expansion and Q-expansion and was described, for example, in [4]. Then Φ is a faithful ball family for packing dimension calculation.
Proof. LetQ 0 be the set ofQ-rational points, and E ′ be any subset of [0; 1]. Let E = E ′ \Q 0 . SinceQ 0 is countable, it follows that The proof is completed by showing that ∆(a, b) be theQ-cylindric interval of the minimal rank such that Denote the rank of ∆(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 ∆(a, b) by ∆ ′ . If the second cylinder exists, then we denote it by ∆ ′′ .
Let us consider the following two cases.
Case 1. The ∆ ′′ does not exist. Then and, therefore, Case 2. The ∆ ′′ exists. Then Summary of the two cases. For every interval (a; b), there exists aQ-cylindric interval ∆(a, b) such that a+b 2 ∈ ∆(a; b) and It follows that the family Φ satisfies the conditions of Theorem 4.1 and is faithful for packing dimension calculation.

Proof of the main result
To prove the main result, we need the following two lemmas.
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 ⊂ [0; 1]. Let us fix any numbers m ∈ N, δ > 0 and consider the following sets: Fix some value m and consider any set W m,δ corresponding to this value. There exists ε > 0 such that |c m | ε for any cylinder c m of rank m. Consider the centered ε-packing of the set W m,δ by intervals E j .
For every interval E j , there exists a cylindric interval ∆(E j ) such that: 3. ∆(E j ) has the minimal possible rank. We denote this rank by i j .
We will say that the cylinder Let us estimate the α-volume of packing of the set E by intervals E j :

This inequality is equivalent to
Let us estimate the expression Therefore, Take the suprema over all possible centered packings {E j } of both parts of the previous inequality: P α ε (W m,δ ) 2P α−δ ε(unc) (W m,δ , Φ). Take the limit as ε → 0: We obtain that . Then for all α < α 0 , the left part is equal to infinity. Thus, for all α < α 0 , the right part is equal to infinity too. It follows that Using the definition of W m,δ , 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 ofQ-expansion cylinders under the condition inf q ij > 0. Let F ξ be a distribution function of a random variable ξ with independent Q-digits. Assume that the following condition holds: where ∆ n (x) is the n-rank cylinder that contains x. Then Φ ′ = F (Φ) is faithful for packing dimension calculation.
Proof. Φ ′ is the family of cylinders for someQ-expansion. Denote this expansion bỹ Q ′ and the corresponding numbers q ij by q ′ ij . It is not clear that condition inf q ′ 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 .
To estimate M , we need the following equation: Dividing the nominator and denominator of the last fraction by ln(q 1j1 q 2j2 . . . q (i−1)ji−1 ), we obtain It follows thatQ ′ satisfies the conditions of Lemma 6.1, and therefore Φ ′ is faithful.
We obtain the following inequality: and this contradicts the assumption that F ξ is PDP . Therefore, we will show that if F ξ is PDP, then dim P (µ ξ ) = 1. The next part of the proof consists of two steps: 1. If dim P (µ ξ ) = 1 and B = 0, then F ξ is PDP ; 2. If dim P (µ ξ ) = 1 and B = 0, then F ξ is not PDP .
Let ε be some positive number such that ε < 1 2 q min . Consider the following sets: Step 1. Let us show that if dim P (µ ξ ) = 1 and B = 0, then F ξ is PDP . Since B = 0, we see that lim k→∞ j∈T k ln p j k ln q min = 0.

Consider the fraction
Split this fraction into three terms. Consider the first term It is easy to prove that where |T + ε,k | is the number of elements in T + ε,k . On the other hand, Also,
Since q min q ij q max and |T + ε,k | k, we have 1 + k · 2ε q min · k ln q max lim k→∞ j∈T + ε,k ln p aj(x)j k j=0 ln q aj (x)j 1 − k · 2ε q min · k ln q max and 1 + 2ε Since ε can be arbitrarily small, it follows that Similarly, Therefore, j∈T ε,k ln p aj (x)j k ln q min |T ε,k | ln( qmin 2 ) k ln q min |T ε,k |(ln(q min ) + ln(1/2)) k ln q min , and the second term tends to zero as k → ∞. Consider the third term j∈T k ln p aj (x)j j ln q aj (x)j .

It can be estimated by
j∈T k ln p j k ln q min , and this value tends to zero as k → ∞ because B = 0. We obtain that Denote by Φ the cylinder family of givenQ-expansion. Denote the image of Φ by Using the Billingsley theorem for packing dimension [12], we have To prove that dim P (E) = dim P (F ξ (E)), it suffices to prove that Φ and Φ ′ are faithful.
Faithfulness of Φ is already proved. Faithfulness of Φ ′ was proved in Lemma 1 and Lemma 2. So, we have that dim P (E) = dim P (F ξ (E)) and F ξ is a PDP -transformation.
Step 2. Let us show that if dim P (µ ξ ) = 1 and B > 0, then F ξ is not PDP . Similarly to step 1, consider the fraction µ(∆ a1a2...a k (x) ) λ(∆ a1a2...a k (x) ) = j∈T + ε,k ln p aj (x)j + j∈T ε,k ln p aj(x)j + j∈T k ln p aj (x)j j ln q aj (x)j 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 → ∞). Consider the third term.
Since the last inequality holds for any δ, it follows that dim P F ξ (L) 1 1 + B , and F ξ is not a PDP -transformation. Let F ξ be the distribution function of a random variable ξ with independent Q *representation. Then F ξ preserves the packing dimension if and only if dim P µ ξ = 1; B = 0. Let F ξ be the distribution function of a random variable ξ with independent s-adic digits. Then F ξ preserves the packing dimension if and only if dim P µ ξ = 1; B = 0.