MSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania MSTA44 10.15559/15-MSTA44 Research Article On packing dimension preservation by distribution functions of random variables with independent Q˜ -digits SlutskyiOleksandrslualexvas@gmail.com Department of Mathematical Analysis and Differential Equations of Dragomanov National Pedagogical University of Ukraine, Ukraine 2015 2312201524371389 16102015 13122015 13122015 © 2015 The Author(s). Published by VTeX2015 Open access article under the CC BY license.

The article is devoted to finding conditions for the packing dimension preservation by distribution functions of random variables with independent 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.

 Packing dimension of a set Hausdorff–Besicovitch dimension of a set faithfulness of fine packing system for packing dimension calculation Q̃-expansion of real numbers packing-dimension-preserving transformations 28A78 28A80 Introduction

Let (M,ρ) be a metric space. Suppose that the Hausdorff–Besicovitch dimension dimH [8] is well defined in (M,ρ) . A transformation f:M→M is called dimension-preserving transformation [13] or DP -transformation if dimH(f(E))=dimH(E),∀E⊂M.

Let G(M,dimH) 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∗ - and  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 Q˜ -digits to be in DP -class. The notion of fine covering system faithfulness for dimH 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 dimH . 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 to Q˜ -expansion under a similar condition.

In 1982, C. Tricot [15] introduced the notion of packing dimension dimP . This dimension is in some sense dual to the Hausdorff–Besicovitch dimension: the definition of dimH of a set F is based on ε-coverings of this figure, but the definition of dimP is based on ε-packings (the countable sets of disjoint open balls Bk(rk,ck) , k∈N , with radii rk⩽ε and centers ck∈F ). The packing dimension has all “good” properties of a fractal dimension, such as the countable stability. Therefore, proving or disproving similar results for dimP is important. For example, we consider the group of packing-dimension-preserving transformations (or PDP -transformations).

The transformation f is said to be a PDP -transformation if ∀E⊂M,dimP(f(E))=dimP(E).

There are a lot of problems with proving of many conjectures for dimP 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 Q˜ -digits to be in PDP -class. Namely, J. Li has proven the following theorem.

Let Fξ be the distribution function of a random variable ξ with independent Q˜ -representation. If infi,jqij=q∗>0 0$]]> and infi,jpij=p∗>0 0$]]>, then Fξ preserves the packing dimension if and only if lim supk→∞h1+h2+⋯+hkb1+b2+⋯+bk=1, where hj=−∑i=1njpijlnpij and bj=−∑i=1njpijlnqij .

In Remark 4.2 at the and of article [9], we read: “The conditions infi,jqij=q∗>0 0$]]> and infi,jpij=p∗>0 0$]]> 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 infi,jpij=p∗>0 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 dimP(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 dimP is replaced by “every ball should have a nonempty intersection with the figure.” We prove that, in some wide class of metric spaces (including Rn ), 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 Q˜ -cylinder system and sufficient conditions for the distribution function of a random variable with independent Q˜ -digits to be in the PDP -class. The corresponding theorem is the main result of the paper.

Let infi,jqij:=qmin . Suppose that qmin>0 0$]]>. Let T:={k:k∈N,pk<qmin2};Tk:=T∩{1,2,…,k};B:=lim supk→∞∑j∈Tkln1pjk.

Let Fξ be the distribution function of a random variable ξ with independent Q˜ -representation. Then Fξ preserves the packing dimension if and only if dimPμξ=1;B=0.

Packing dimension

Let us recall the definition of packing dimension in the form given, for example, in [8].

Let E⊂M and ε>0 0$]]>. A finite or countable family {Ej} of open balls is called an ε-packing of a set E if

|Ei|⩽ε for all i;

ci∈E , i∈N , where ci is the center of the ball Ei ;

Ei∩Ej=∅ for all i, j, i≠j .

The empty set of balls is a packing of any set.

Let E⊂M , α⩾0 , ε>0 0$]]>. Then the α-dimensional packing premeasure of a bounded set E is defined by Pεα(E):=sup{∑i|Ei|α},

where the supremum is taken over all at most countable ε-packings {Ej} of E (if Ej=∅ for all j, then Pεα(E)=0 ).

The α-dimensional packing quasi-measure of a set E is defined by P0α(E):=limε→0Pεα(E).

The α-dimensional packing measure is defined by Pα(E):=inf{∑jP0α(Ej):E⊂⋃Ej},

where the infimum is taken over all at most countable coverings {Ej} of E, Ej⊂M .

The nonnegative number dimP(E):=inf{α:Pα(E)=0} is called the uncentered packing dimension of a set E⊂M .

 Uncentered packing dimension

Let E⊂M and ε>0 0$]]>. A finite or countable family {Ej} of open balls is called an uncentered ε-packing of a set E if

|Ei|⩽ε for all i;

Ei∩E≠∅ ;

Ei∩Ej=∅ for all i, j, i≠j .

The empty set of balls is an uncentered packing of any set.

Let E⊂M , α⩾0 , ε>0 0$]]>. Then the uncentered α-dimensional packing premeasure of a bounded set E is defined by Pε(unc)α(E):=sup{∑i|Ei|α},

where the supremum is taken over all at most countable uncentered ε-packings {Ei} of E.

The uncentered α-dimensional packing quasi-measure of a set E is defined by P0(unc)α(E):=limε→0Pε(unc)α(E).

Uncentered α-dimensional packing measure is defined by P(unc)α(E):=inf{∑jP0(unc)α(Ej):E⊂⋃Ej},

where the infimum is taken over all at most countable coverings {Ej} of E, Ej⊂M .

If (M,ρ)=R1 and α=1 , then the α-dimensional packing measure and uncentered α-dimensional packing measure are the Lebesgue measure.

The nonnegative number dimP(unc)(E):=inf{α:P(unc)α(E)=0}. is called the uncentered packing dimension of a set E⊂M .

Let (M,ρ) be a metric space. Let C∈N . If for all r>0 0$]]> and for any open ball I with |I|=8r , there exist at most N(I) balls Ii , i∈{1,…,N(I)} such that Ii⊂I , i∈{1,…,N(I) , |Ii|=r , i∈{1,…,N(I)} , and N(I)≤C . Then dimP(unc)(E)=dimP(E).

Step 1. Let us prove the inequality dimP(unc)(E)⩾dimP(E) .

By the definitions and supremum property we have Pr(unc)α(E)⩾Prα(E).

By the limit property of inequalities we have P0(unc)α(E)⩾P0α(E).

Hence, P(unc)α(E)⩾Pα(E). Let dimP(unc)(E)=α0 . By the definition of dimP(unc)(E) we have ∀ε>0,P(unc)α0+ε(E)=0. 0,\hspace{1em}{\mathcal{P}_{(\mathit{unc})}^{\alpha _{0}+\varepsilon }}(E)=0.\]]]> Therefore, ∀ε>0,P0α0+ε(E)=0, 0,\hspace{1em}{\mathcal{P}_{0}^{\alpha _{0}+\varepsilon }}(E)=0,\]]]> and, consequently, dimP(E)⩽α0. Hence, it follows that dimP(unc)(E)⩾dimP(E) , which is our claim.

Step 2. Let us show that dimP(unc)(E)⩽dimP(E) .

If dimP(unc)(E)=0 , then the statement is true.

Let us consider the case dimP(unc)(E)≠0 . Fix 0<t<s<dimP(unc)(E) .

Since s<dimP(unc)(E), we have P(unc)s(E)=+∞,P0(unc)s(E)=+∞. Therefore, ∀r>0,Pr(unc)s(E)=+∞. 0,\hspace{2.5pt}{\mathcal{P}_{r(\mathit{unc})}^{s}}(E)=+\infty .\]]]> From this and from the supremum property, it follows that there is an uncentered packing V:={Ei} of the set E with ∑i|Ei|s>1. 1.\]]]> Let us divide the packing V into classes Vk:={Ei:2−k−1⩽|Ei|<2−k}. Let nk be the number of balls Vk . We will show that ∃k0:nk0⩾2k0t(1−2t−s).

To obtain a contradiction, suppose that nk<2kt(1−2t−s)for allk. Then ∑i|Ei|s<∑k2−ks·nk<∑k2−ks·2kt(1−2t−s)=(1−2t−s)·∑k(2t−s)k=1, which contradicts our assumption (1).

Therefore, such k0 exists. Let us consider Vk0 . We denote by A1,A2,…,Ank0 the balls in Vk0 , that is, Vk0={A1,A2,…,Ank0}.

Fix r:=2−k0−1 . Then the radius of any Ai is less than r. Let Ti be a point of Ai such that Ti∈Ai∩E . Let V′ be the set of balls with the centers Ti and radius r, that is, V′={Ai′:Ai′=B(Ti,r)}. Fix V∗={Ai∗:Ai∗=B(Ti,4r)}.

Let us divide the set V′ into classes K1,K2,…,Kl as follows.

Let us take a ball Aj1′=A1′ and put it in K1 together with all other balls Ai′∈V′ such that Ai′∩Aj1′≠∅ .

Let us take an arbitrary ball Aj2′∈V′∖K1 and put it in K2 together with all other balls Ai′∈V′∖K1 such that Ai′∩Aj2′≠∅ .

Let us continue this way until V′∖(K1∪K2∪⋯∪Kl)≠∅ . Since the number of elements in a set V′ is a finite, we can find such a number l.

Now suppose that the balls Ai′ and Aj′ intersect each other. In other words, ρ(Ti,Tj)⩽2r . Therefore, Aj⊂Ai∗ .

The radius of Aj 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 Ki .

Moreover, in the case i<m , the balls Aji′ and Ajm′ do not intersect each other. Indeed, suppose otherwise. Then Ajm′ is in a class Ki or in a class with number less than i.

Hence, V″={Aj1′,Aj2′,…,Ajl′} 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 Vk0 no more than C times. Therefore, ∑V″|Aji′|t⩾nk0·2−k0tC⩾2k0t(1−2t−s)·2−k0tC=1−2t−sC. From this it follows that P2−k0t(E)⩾1−2t−sC.

By the inequality 2−k0<r we get Prt(E)⩾1−2t−sCfor allr>0. 0.\]]]>

Consequently, as r→0 , we get the inequality P0t(E)⩾1−2t−sC.

Let us show that Pt(E)⩾1−2t−sC . Recall the definition Pt(E)=inf{∑jP0t(Ej):E⊂⋃Ej}, where the infimum is taken over all at most countable coverings Ej of a set E.

Let {Ej} be an at most countable covering of E. Since dimP(unc)(E)>s s$]]>, there is j0 such that dimP(unc)(Ej0)>s s$]]> (by the countable stability of the packing dimension dimP(unc) ). In other words, we have P(unc)s(Ej0)=+∞,P0(unc)s(Ej0)=+∞. We conclude by the part of the theorem already proved for E that P0t(Ej0)⩾1−2t−sC and ∑jP0t(Ej)⩾1−2t−sC. But the previous inequality is true for an arbitrary covering {Ej} of a set E and for the infimum for all coverings. Therefore, Pt(E)⩾1−2t−sC and dimP(E)⩾t. Since t-dimP(unc)(E) can be approximated by 0, we get dimP(E)⩾dimP(unc)(E) , which completes the proof.  □

If M=Rn , then dimP(unc)(E)=dimP(E) .

Let B8r be a ball with radius 8r , Br be a ball with radius r, and λ be the n-dimensional Lebesgue measure. Then λ(B8r)=8n·λ(Br).

Therefore, we can put no more than C=8n 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) .

Let E⊂M , α⩾0 , ε>0 0$]]>. Then the α-dimensional packing premeasure of a bounded set E with respect to Φ is defined by Pεα(E,Φ):=sup{∑i|Ei|α},

where the supremum is taken over all uncentered ε-packings {Ei}⊂Φ of E (if {Ei}=∅ , then Pεα(E,Φ)=0 ).

The α-dimensional packing quasi-measure of a set E w.r.t. Φ is defined by P0α(E,Φ):=limε→0Pεα(E,Φ).

The α-dimensional packing measure w.r.t. Φ is defined by Pα(E,Φ):=inf{∑jP0α(Ej,Φ):E⊂⋃Ej},

where the infimum is taken over all at most countable coverings {Ej} of E, Ej⊂M .

The nonnegative number dimP(E,Φ):=inf{α:Pα(E,Φ)=0} is called the packing dimension of a set E⊂M w.r.t. Φ.

In the definition of dimP(E,Φ) , we used uncentered packing. But we will denote this dimension without index (unc) because:

We will work in Rn . In this space, centered and uncentered packing dimensions are equal;

The centered packing dimension w.r.t. some family of balls is not defined.

dimP(E,Φ)⩽dimP(unc)(E).

Let Φ0 be the family of all open balls of M. Then Pr(unc)α(E)=Prα(E,Φ0).

Since Φ⊆Φ0 , by the supremum property we have Prα(E,Φ)⩽Prα(E,Φ0).

By the inequality for packing premeasures it follows that dimP(E,Φ)⩽dimP(unc)(E), which proves the theorem.  □

 Faithfulness of the open balls families for packing dimension calculation

Suppose that some open balls family Φ satisfies the following condition: for all E⊂M , dimP(unc)(E,Φ)=dimP(unc)(E). Then Φ is said to be faithful for uncentered packing dimension calculation.

The notion of faithfulness is introduced for the Hausdorff–Besicovitch dimension dimH [11]. It is clear that ∀Φ⊂2M,dimH(E,Φ)⩾dimH(E).

(The sufficient condition for the open-ball family to be faithful for packing dimension calculation).

Suppose that

Φ is a family of intervals from [0;1] ;

∃C>0:∀(a;b)⊂[0;1] 0:\forall (a;b)\subset [0;1]$]]>, ∃Δ(a;b)∈Φ such that:

a+b2∈Δ(a,b) ;

Δ(a,b)⊂(a;b) ;

b−a|Δ(a,b)|⩾C .

 Then Φ is a faithful open-ball family for packing dimension calculation.

Let E be any set, α⩾0 , and r>0 0$]]>. Let {Ei}={(ai;bi)} be a family of disjoint intervals such that ai+bi2∈E and bi−ai<r .

Then the following inequality holds: ∑i|Ei|α⩽∑i|Δ(ai,bi)|α·Cα.

Taking the supremum (over all sets of intervals {Ei} satisfying the previous conditions), we have Prα(E)⩽sup{Ei}|Δ(ai,bi)|α·Cα.

Any set of intervals {Δ(ai,bi)} satisfies the conditions from the Pr(unc)α(E,Φ) definition. So, sup{Ei}|Δ(ai,bi)|α·Cα⩽Pr(unc)α(E,Φ)·Cα. Therefore, Prα(E)⩽Pr(unc)α(E,Φ)·Cα. Taking the limit of both sides, we have P0α(E)⩽P0(unc)α(E,Φ)·Cα.

Taking the infimum over all possible coverings of the set E, we have Pα(E)⩽P(unc)α(E,Φ)·Cα and dimP(E)⩽dimP(unc)(E,Φ). Since [0;1]⊂R1 , it follows that dimP(E)=dimP(unc)(E) and dimP(unc)(E)⩽dimP(unc)(E,Φ). Using dimP(unc)(E)⩾dimP(unc)(E,Φ)for allΦ, we obtain that Φ is a faithful open-ball family for the packing dimension calculation.  □

 Sufficient conditions for <inline-formula id="j_vmsta44_ineq_205"><alternatives> <mml:math><mml:mover accent="true"><mml:mrow><mml:mi mathvariant="italic">Q</mml:mi></mml:mrow><mml:mo stretchy="false">˜</mml:mo></mml:mover></mml:math> <tex-math><![CDATA[$\tilde{Q}$]]></tex-math></alternatives></inline-formula>-expansion cylindric interval family to be faithful

The Q˜ -expansion of real numbers is a generalization of s-expansion and Q-expansion and was described, for example, in [4].

Let Φ be the system of cylindric intervals of some Q˜ -expansion. Suppose that infi,jqij=qmin>0. 0.\]]]> Then Φ is a faithful ball family for packing dimension calculation.

Let Q˜0 be the set of Q˜ -rational points, and E′ be any subset of [0;1] . Let E=E′∖Q˜0 . Since Q˜0 is countable, it follows that dimP(unc)(Q˜0)=0,dimP(unc)(Q˜0,Φ)=0, and dimP(unc)(E′)=dimP(unc)(E),dimP(unc)(E′,Φ)=dimP(unc)(E,Φ). The proof is completed by showing that dimP(unc)(E)=dimP(unc)(E,Φ) for every set E⊂[0;1] if E does not contain Q˜ -rational points.

Let (a;b)⊂[0;1] . Let Δ(a,b) be the Q˜ -cylindric interval of the minimal rank such that a+b2∈Δ(a;b)⊂(a;b).

Denote the rank of Δ(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 Δ(a,b) by Δ′ . If the second cylinder exists, then we denote it by Δ″ .

Let us consider the following two cases.

The Δ″ does not exist. Then |Δ(a,b)|⩽b−a⩽|Δ′|, and, therefore, |Δ(a,b)|⩾(b−a)·qmin.

The Δ″ exists. Then |Δ′|·2⩾b−a⇒|Δ(a,b)|⩾(b−a)·qmin2.

Summary of the two cases. For every interval (a;b) , there exists a Q˜ -cylindric interval Δ(a,b) such that a+b2∈Δ(a;b) and |Δ(a,b)|⩾(b−a)·qmin2. It follows that the family Φ satisfies the conditions of Theorem 4.1 and is faithful for packing dimension calculation.  □

Let Φ be a family of Q∗ -cylinders under the condition infi,jqij>0 0$]]>. Then Φ is faithful for packing dimension calculation.

Let Φ be a family of Q-cylinders. Then Φ is faithful for packing dimension calculation.

Let Φ be a family of s-adic cylinders. Then Φ is faithful for packing dimension calculation.

 Proof of the main result

To prove the main result, we need the following two lemmas.

Let Q˜ be the matrix ‖qik‖ , i∈N , k∈{0,1,…,Nk−1} . If limi→∞lnqikkln(qi11qi22…qik−1(k−1))=0 for every sequence (ik) , then the open-ball family Φ of the respective expansion cylinder interiors is faithful for packing dimension calculation.

Let us fix a set E⊂[0;1] . Let us fix any numbers m∈N , δ>0 0$]]> and consider the following sets: Wm,δ={x∈E:lnqikk(x)ln(qi11(x)qi22(x)…qik−1(k−1)(x))<δ,∀k⩾m}.

Fix some value m and consider any set Wm,δ corresponding to this value. There exists ε>0 0$]]> such that |cm|⩾ε for any cylinder cm of rank m. Consider the centered ε-packing of the set Wm,δ by intervals Ej .

For every interval Ej , there exists a cylindric interval Δ(Ej) such that:

Δ(Ej)⊂Ej ;

Δ(Ej) contains the middle point xj of the Ej ;

Δ(Ej) has the minimal possible rank. We denote this rank by ij .

We will say that the cylinder Δ′(Ej) is the “father” of Δ(Ej) if Δ′(Ej)⊃Δ(Ej) and the rank of Δ′(Ej) is equal to ij−1 . It is obvious that |Δ′(Ej)|⩾|Ej|2 . Therefore, |Ej|⩽2|Δ(Ej)|qikjkj(xj),wherexj∈Wm,δ.

Let us estimate the α-volume of packing of the set E by intervals Ej : ∑k|Ej|α⩽∑k|Δ(Ej)|α·(2qikjkj(xj))α.

This inequality is equivalent to ∑k|Ej|α⩽∑k|Δ(Ej)|α−δ·|Δ(Ej)|δ·(2qikjkj(xj))α. Let us estimate the expression ln(|Δ(Ej)|δ·(2qikjkj(xj))α)=δln(qi11(x)qi22(x)…qikj−1(kj−1)(x))+αln2−αlnqikjkj(xj). Since xj∈Wm,δ , it follows that δln(qi11(x)qi22(x)…qikj−1(kj−1))⩽lnqikjkj(xj). Therefore, ln(|Δ(Ej)|δ·(2qikjkj(xj))α)⩽αln2+(1−α)lnqikjkj(xj)⩽αln2. Thus, we have ∑j|Ej|α⩽2∑j|Δ(Ej)|α−δ. Take the suprema over all possible centered packings {Ej} of both parts of the previous inequality: Pεα(Wm,δ)⩽2Pε(unc)α−δ(Wm,δ,Φ). Take the limit as ε→0 : P0α(Wm,δ)⩽2P0(unc)α−δ(Wm,δ,Φ). We obtain that Pα(Wm,δ)⩽2P(unc)α−δ(Wm,δ,Φ). Denote α0=dimP(Wm,δ) . Then for all α<α0 , the left part is equal to infinity. Thus, for all α<α0 , the right part is equal to infinity too. It follows that dimP(unc)(Wm,δ,Φ)⩾α−δ and dimP(unc)(Wm,δ,Φ)⩾dimP(Wm,δ)−δ. Using the definition of Wm,δ , we get E= ⋃m=1∞Wm,δ. Now, by packing dimension countable stability, dimP(unc)(E,Φ)⩾dimP(E)−δ. Since δ can be arbitrarily small, dimP(unc)(E,Φ)⩾dimP(E). To complete the proof, it remains to note that E is any subset of [0;1] . Thus, Φ is faithful.  □

Let Φ be a family of Q˜ -expansion cylinders under the condition infqij>0 0$]]>. Let Fξ be a distribution function of a random variable ξ with independent Q˜ -digits. Assume that the following condition holds: limn→∞lnλ(F(Δn(x)))lnλ(Δn(x))=1,∀x∈[0;1], where Δn(x) is the n-rank cylinder that contains x.

Then Φ′=F(Φ) is faithful for packing dimension calculation.

Φ′ is the family of cylinders for some Q˜ -expansion. Denote this expansion by Q˜′ and the corresponding numbers qij by qij′ .

It is not clear that condition infqij′>0 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(Δa1a2…anQ˜(x))=Δa1a2…anQ˜′(x) and lnλ(F(Δn(x)))=ln(q1a1′q2a2′…qnan′). Denote M=lim supi→∞lnqiji′ln(q1j1′q2j2′…q(i−1)ji−1′). To estimate M, we need the following equation: limn→0lnλ(F(Δn(x)))lnλ(Δn(x))=limn→0ln(q1j1′q2j2′…q(i−1)ji−1′)+lnqiji′ln(q1j1q2j2…q(i−1)ji−1)+lnqiji. Dividing the nominator and denominator of the last fraction by ln(q1j1q2j2…q(i−1)ji−1) , we obtain limn→01+lnqiji′ln(q1j1′q2j2′…q(i−1)ji−1′)ln(q1j1q2j2…q(i−1)ji−1)ln(q1j1′q2j2′…q(i−1)ji−1′)+lnqijiln(q1j1′q2j2′…q(i−1)ji−1′)=1+M1+0=1⇒M=0. It follows that Q˜′ satisfies the conditions of Lemma 6.1, and therefore Φ′ is faithful.  □

Let us show that if Fξ is PDP , then dimP(μξ)=1 .

Assume the converse. Then there exists a set Eα such that μξ(Eα)=1 and dimP(Eα)=α . Consider Fξ(Eα) . Since μξ(Eα)=1 , we have λ(Eα)=1 , and thus dimP(Fξ(Eα))=1 .

We obtain the following inequality: dimP(Fξ(Eα))=1≠α=dimP(Eα), and this contradicts the assumption that Fξ is PDP . Therefore, we will show that if Fξ is PDP , then dimP(μξ)=1 .

The next part of the proof consists of two steps:

If dimP(μξ)=1 and B=0 , then Fξ is PDP ;

If dimP(μξ)=1 and B≠0 , then Fξ is not PDP .

Let ε be some positive number such that ε<12qmin . Consider the following sets: Tε,k+={j:j∈N,j⩽k,|pij−qij|⩽ε,i∈{0,1,…,s−1}},Tε,k−={1,2,…,k}∖Tε,k+,T={k:k∈N,pk<12qmin},Tk=T∩{1,2,…,k},Tε,k=Tε,k−∖Tk.

Step 1. Let us show that if dimP(μξ)=1 and B=0 , then Fξ is PDP . Since B=0 , we see that limk→∞∑j∈Tklnpjklnqmin=0.

Consider the fraction lnμξ(Δa1a2…ak(x))lnλ(Δa1a2…ak(x))=∑j∈Tε,k+lnpaj(x)j+∑j∈Tε,klnpaj(x)j+∑j∈Tklnpaj(x)j∑jlnqaj(x)j.

Split this fraction into three terms. Consider the first term ∑j∈Tε,k+lnpaj(x)j∑jlnqaj(x)j.

It is easy to prove that ∑j∈Tε,k+lnpaj(x)j⩾∑j∈Tε,k+ln(qaj(x)j−ε)=∑j∈Tε,k+(lnqaj(x)j+ln(qaj(x)j−εqaj(x)j))⩾∑j∈Tε,k+(lnqaj(x)j)+|Tε,k+|·2εqmin, where |Tε,k+| is the number of elements in Tε,k+ . On the other hand, ∑j∈Tε,k+lnpaj(x)j⩽∑j∈Tε,k+(lnqaj(x)j)−|Tε,k+|·2εqmin. Also, 1+|Tε,k+|·2εqmin·∑j=0klnqaj(x)j⩽limk→∞∑j∈Tε,k+lnpaj(x)j∑j=0klnqaj(x)j⩽1−|Tε,k+|·2εqmin·∑j=0klnqaj(x)j (note that |Tε,k+|·2εqmin·∑j=0klnqaj(x)j<0 ).

Since qmin⩽qij⩽qmax and |Tε,k+|⩽k , we have 1+k·2εqmin·klnqmax⩽limk→∞∑j∈Tε,k+lnpaj(x)j∑j=0klnqaj(x)j⩽1−k·2εqmin·klnqmax and 1+2εqmin·lnqmax⩽limk→∞∑j∈Tε,k+lnpaj(x)j∑j=0klnqaj(x)j⩽1−2εqmin·lnqmax. Since ε can be arbitrarily small, it follows that limk→∞∑j∈Tε,k+lnpaj(x)j∑j=0klnqaj(x)j=1.

Similarly, |Tε,k|ln(qmin2)⩽∑j∈Tε,klnpaj(x)j⩽|Tε,k|ln(2−qmin2).

Therefore, ∑j∈Tε,klnpaj(x)jklnqmin⩽|Tε,k|ln(qmin2)klnqmin⩽|Tε,k|(ln(qmin)+ln(1/2))klnqmin, and the second term tends to zero as k→∞ .

Consider the third term ∑j∈Tklnpaj(x)j∑jlnqaj(x)j. It can be estimated by ∑j∈Tklnpjklnqmin, and this value tends to zero as k→∞ because B=0 .

We obtain that limk→∞μξ(Δa1a2…ak(x))λ(Δa1a2…ak(x))=1.

Denote by Φ the cylinder family of given Q˜ -expansion. Denote the image of Φ by Φ′=Fξ(Φ) .

Using the Billingsley theorem for packing dimension [12], we have dimP(E,Φ)=1·dimP(Fξ(E),Φ′)∀E⊂[0;1].

To prove that dimP(E)=dimP(Fξ(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 dimP(E)=dimP(Fξ(E)) and Fξ is a PDP -transformation.

Step 2. Let us show that if dimP(μξ)=1 and B>0 0$]]>, then Fξ is not PDP .

Similarly to step 1, consider the fraction μ(Δa1a2…ak(x))λ(Δa1a2…ak(x))=∑j∈Tε,k+lnpaj(x)j+∑j∈Tε,klnpaj(x)j+∑j∈Tklnpaj(x)j∑jlnqaj(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 B>0 0$]]>, there exists a subsequence (km) such that limm→∞∑j∈Tkmln1pjkm=B. Consider the set L=x:x=Δa1a2…ak…;{ak∈{0,1,…,s−1}ifk∉Tak=nk,ifk∈T,wherepnkk=minipik. Since the digits are in infinitely many places, it follows that λ(L)=0 . But combining limm→∞|Tkm|km=0 and the formula for dimP(μξ) , we have dimP(L)=1 . It follows that ∀x∈Llimm→∞lnμ(Δa1a2…akm(x))lnλ(Δa1a2…akm(x))=1+B. Thus, for any δ>0 0$]]>, there exists m(δ) such that for all m>m(δ) m(\delta )$]]>, we have 1+B−δ⩽lnμ(Δa1a2…akm(x))lnλ(Δa1a2…akm(x))⩽1+B+δ. Thus, we have lim infk→∞lnμ(Δa1a2…ak(x))lnλ(Δa1a2…ak(x))⩾1+B−δ, and (using the Billingsley theorem for dimP ) dimP−μ(L)·(1+B−δ)⩽dimP(L), that is, dimP(Fξ(L))⩽11+B−δ. Since the last inequality holds for any δ, it follows that dimP(Fξ(L))⩽11+B, and Fξ is not a PDP -transformation.  □

Let infi,jqij:=qmin . Suppose that qmin>0 0$]]>. Let T:={k:k∈N,pk<qmin2};Tk:=T∩{1,2,…,k};B:=lim supk→∞∑j∈Tkln1pjk.

Let Fξ be the distribution function of a random variable ξ with independent Q∗ -representation. Then Fξ preserves the packing dimension if and only if dimPμξ=1;B=0.

Let s∈N , s⩾2 ; T:={k:k∈N,pk<12s};Tk:=T∩{1,2,…,k};B:=lim supk→∞∑j∈Tkln1pjk.

Let Fξ be the distribution function of a random variable ξ with independent s-adic digits. Then Fξ preserves the packing dimension if and only if dimPμξ=1;B=0.

References Albeverio, S., Torbin, G.: Fractal properties of singularly continuous probability distributions with independent q∗ -digits. Bull. Sci. Math. 129(4), 356–367 (2005). MR2134126. doi:10.1016/j.bulsci.2004.12.001 Albeverio, S., Pratsiovytyi, M., Torbin, G.: Fractal probability distributions and transformations preserving the Hausdorff–Besicovitch dimension. Ergod. Theory Dyn. Syst. 24(1), 1–16 (2004). MR2041258. doi:10.1017/S0143385703000397 Albeverio, S., Pratsiovytyi, M., Torbin, G.: Transformations preserving the Hausdorff–Besicovitch dimension. Cent. Eur. J. Math. 6(1), 119–128 (2008). MR2379954. doi:10.2478/s11533-008-0007-y Albeverio, S., Koshmanenko, V., Pratsiovytyi, M., Torbin, G.: On fine structure of singularly continuous probability measures and random variables with independent Q˜ -symbols. Methods Funct. Anal. Topol. 2(1), 97–111 (2011). MR2849470 Albeverio, S., Ivanenko, G., Lebid, M., Torbin, G.: On the Hausdorff dimension faithfulness and the Cantor series expansion. arXiv:1305.6036 Billingsley, P.: Hausdorff dimension in probability theory. II. Ill. J. Math. 5(1), 291–298 (1961). MR0120339 Das, M.: Billingsley’s packing dimension. Proc. Am. Math. Soc. 136(1), 273–278 (2008). MR2350413. doi:10.1090/S0002-9939-07-09069-7 Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester (2003). MR2118797. doi:10.1002/0470013850 Li, J.: A class of probability distribution functions preserving the packing dimension. Stat. Probab. Lett. 81(1), 1782–1791 (2011). MR2845889. doi:10.1016/j.spl.2011.07.010 Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge (1995). MR1333890. doi:10.1017/CBO9780511623813 Nikiforov, R., Torbin, G.: Fractal properties of random variables with independent Q∞ -digits. Probab. Theory Math. Stat. 86, 150–162 (2012). MR2986457. doi:10.1090/S0094-9000-2013-00896-5 Slutskyi, O., Torbin, G.: Analog of the Billingsley theorem for the packing dimension. Trans. Natl. Pedagog. Univ. Ukr., Math. 1, 192–199 (2012). Torbin, G.: Probabilistic approach to transformations saving fractal dimension. Math. Bull. Shevchenko Sci. Soc. 4(1), 275–283 (2007) (in Ukrainian) Torbin, G.: Probability distributions with independent q-symbols and transformations preserving the Hausdorff dimension. Theory Stoch. Process. 13(1), 281–293 (2007). MR2343830 Tricot, C.: Two definitions of fractional dimension. Math. Proc. Camb. Philos. Soc. 91(1), 57–74 (1982). MR0633256. doi:10.1017/S0305004100059119 Turbin, A., Pratsiovytyi, N.: Fractal Sets, Functions, and Distributions, Naukova Dumka, Kyiv (1992) (in Russian). MR1353239