On the Hausdorff dimension faithfulness and the Cantor series expansion

We study families $\Phi$ of coverings which are faithful for the Hausdorff dimension calculation on a given set $E$ (i. e., special relatively narrow families of coverings leading to the classical Hausdorff dimension of an arbitrary subset of $E$) and which are natural generalizations of comparable net-coverings. They are shown to be very useful for the determination or estimation of the Hausdorff dimension of sets and probability measures. We give general necessary and sufficient conditions for a covering family to be faithful and new techniques for proving faithfulness/non-faithfulness for the family of cylinders generated by expansions of real numbers. Motivated by applications in the multifractal analysis of infinite Bernoulli convolutions, we study in details the Cantor series expansion and prove necessary and sufficient conditions for the corresponding net-coverings to be faithful. To the best of our knowledge this is the first known sharp condition of the faithfulness for a class of covering families containing both faithful and non-faithful ones. Applying our results, we characterize fine fractal properties of probability measures with independent digits of the Cantor series expansion and show that a class of faithful net-coverings essentially wider that the class of comparable ones. We construct, in particular, rather simple examples of faithful families $\mathcal{A}$ of net-coverings which are"extremely non-comparable"to the Hausdorff measure.


Introduction
The notion of the Hausdorff dimension is well-known now and is of great importance in mathematics as well as in diverse applied problems (see, e.g., [12,23,28,36,45]). In many situations the determination (or even estimations) of this dimension for sets from a given family or even for a given set is a rather non-trivial problem (see, e. g., [1,8,13,17,23,32,37] and references therein). Different approaches and special methods for the determination of the Hausdorff dimension are collected in [23,24,31]. A new approach based on the theory of transformations preserving the Hausdorff dimension (DP-transformations) was presented in [7,9] and has been developed in [4,26,27]. In this paper we develop another approach which is deeply connected with the theory of DP-transformations as well as with the following well known approach: to simplify the calculation of the Hausdorff dimension of a given set it is extremely useful to have an appropriate and a relatively narrow family of admissible coverings which lead to the same value of the dimension. We shall start to deal with one-dimensional sets and show later how our results can be naturally extended to the multidiminsional case and to the general case of metric spaces.
Without loss of generality we shall consider subsets from the unit interval. Let \Phi be a fine family of coverings on [0, 1], i.e., a family of subsets of [0, 1] such that for any \varepsi > 0 there exists an at most countable \varepsi -covering \{ E j \} of [0, 1] with E j \in \Phi . Let where the infimum is taken over all at most countable \varepsi -coverings \{ E j \} of E, E j \in \Phi . We remark that, generally speaking, H \alpha (E, \Phi ) depends on the family \Phi . The notion of comparable net measures are also well known now (see, e. g. , [22,40]). Roughly speaking, net measures are special cases of H \alpha (E, \Phi ), where the family \Phi consists of sets with the following properties: 1) if A 1 and A 2 belong to \Phi , then A 1 \subset A 2 or at most a finite number of sets from \Phi contain any given set from \Phi . Then the corresponding net measure H \alpha (E, \Phi ) is said to be comparable to Hausdorff measure if the ratios of measures are bounded above and below. It has been shown that comparable net measures are very useful in the study of Hausdorff measures (see, e.g., [15,22,30,40] and references therein).
In this paper we actually develop theory of measures which are generalizations of comparable net measures in the following sense. Definition. A fine covering family \Phi is said to be faithful family of coverings (non-faithful family of coverings) for the Hausdorff dimension calculation on [0, 1] if \mathrm{ \mathrm{ \mathrm{ H (E, \Phi ) = \mathrm{ \mathrm{ \mathrm{ H (E), \forall E \subsete [0, 1] (resp. \exists E \subsete [0, 1] : \mathrm{ \mathrm{ \mathrm{ H (E, \Phi ) \not = \mathrm{ \mathrm{ \mathrm{ H (E)). It is clear that any family \Phi of comparable net-coverings (i.e., net-coverings which generate comparable net-measures) is faithful. Conditions for a fine covering family to be faithful were studied by many authors (see, e.g., [11,16,20,39] and references therein). First steps in this direction have been done by A. Besicovitch ( [15]), who proved the faithfulness for the family of cylinders of binary expansion. His result was extended by P. Billingsley ([16]) to the family of s-adic cylinders, by M. Pratsiovytyi ([44]) to the family of Q-S-cylinders, and by S. Albeverio and G. Torbin ([11]) to the family of Q \ast -cylinders for those matrices Q \ast whose elements p 0k , p (s - 1)k are bounded from zero. Some general sufficient conditions for the faithfulness of a given family of coverings are also known ( [20], [39]). Let us mentioned here that all these results were obtained by using the standard approach: if for a given family \Phi there exist positive constants \beta \in \BbbR and N \ast \in \BbbN such that for any interval B = (a, b) there exist at most N \ast sets B j \in \Phi which cover (a, b) and | B j | \leq \beta \cdot | B| , then the family \Phi is faithful. It is clear that all above mentioned families of net-coverings are even comparable. The faithfulness of underlying families of basic cylinders plays a crucial role in the studying of fractal properties of non-normal as well as essentially non-normal numbers in different systems of numerations (see, e.g., [1,4,5,6,38]) and fine fractal properties of singularly continuous probability measures (see, e.g., [1,11,32] and references therein).
It is rather paradoxical that initial examples of non-faithful families of coverings appeared firstly in two-dimensional case (as a result of active studies of self-affine sets during the last decade of XX century (see, e.g., [14])). The family of cylinders of the classical continued fraction expansion can probably be considered as the first (and rather unexpected) example of non-faithful one-dimensional net-family of coverings ( [35]). By using approach, which has been invented by Yuval Peres to prove non-faithfulness of the family of continued fraction cylinders ( [35]), in [2] authors have proven the non-faithfulness for the family of cylinders of Q \infty -expansion with polynomially decreasing elements \{ q i \} . The latter two families of coverings give examples of non-comparable net measures.
So, it is natural to ask about the existence of faithful covering families which are not comparable.
We study this problem and give general necessary and sufficient conditions for a fine covering system to be faithful. The main aim of the paper is to study faithful properties of the covering families which are generated by the famous Cantor series expansions. Let us recall that for a given sequence \{ n k \} \infty k=1 with n k \in \BbbN \setminu \{ 1\} , k \in \BbbN the expression of x \in [0, 1] in the following form x = \infty \sum k=1 \alpha k n 1 \cdot n 2 \cdot . . . \cdot n k =: \Delta \alpha 1\alpha 2...\alpha k ... , \alpha k \in \{ 0, 1, ..., n k -1\} is said to be the Cantor series expansion of x. These expansions, which have been initially studied by G. Cantor in 1869 (see., e.g. [18]), are natural generalizations of the classical s-adic expansion for reals. Cantor series expansions have been intensively studied from different points of view during last century (see, e.g., [21,29,41,25] and references therein). Our own motivations to study faithful properties of such expansions came from our investigations on fine fractal properties of infinite Bernoulli convolutions, i.e., probability distributions of the following random variables where \infty \sum k=1 a k is a convergent positive series, and \xi k are independent random variables taking values 0 and 1 with probabilities p 0k and p 1k respectively. Measures of this form have been studied since 1930's from the pure probabilistic point of view as well as for their applications in harmonic analysis, dynamical systems and fractal analysis [33]. The Lebesgue structure and fine fractal properties of the distribution of \xi are well studied for the case where r k := \infty \sum i=k+1 a i \geq a k for all large enough k (see, e.g., [10,19] case where a k < r k holds for an infinite number of k can be considered as a «Terra incognita» in this field. Even for the case where a k = \lambda k and p 0k = 1 2 the problem of singularity is still open ( [42]). Main problems here are related to the fact that almost all points from the spectrum of \xi have uncountably many different expansion in the form \sum \varepsi k a k , \varepsi k \in \{ 0, 1\} . This is the so-called «Bernoulli convolutions with large overlaps». We consider two special classes of such measures. The first one is generated by sequences a k with the following properties: The second one is connected to the sequences a k such that \forall k \in N \exists s k \in N \bigcup \{ 0\} : a k = a k+1 = ... = a k+s k = r k+s k , and s k > 0 for an infinite number of indices k. In both cases singularity plays a generic role and to study fine fractal properties of the corresponding probability distributions it is necessary to have knowledge on faithfulness (non-faithfulness) of fine families of partitions which are turned to be the Cantor series partitions.
Main result of the present paper states that the family \scrA of Cantor coverings of the unit interval is faithful for the Hausdorff dimension calculation if and only if \mathrm{ \mathrm{ \mathrm{ k\rightar\infty \mathrm{ \mathrm{ n k \mathrm{ \mathrm{ n 1 \cdot n 2 \cdot . . . \cdot n k - To the best of our knowledge this theorem gives the first necessary and sufficient condition of the faithfulness for a class of covering families containing both faithful and non-faithful ones. The proof of this result is given in the next Section. As a corollary of our results we characterize fine fractal properties of probability measures with independent digits of the Cantor series expansion and show that a class of faithful net-coverings essentially wider that the class of comparable net-coverings. We construct, in particular, simple examples of faithful families \scrA of net-coverings which are "extremely" non-comparable to the Hausdorff measure.

Sharp conditions for the Hausdorff dimension faithfulness of the Cantor series expansion
In this Section we give some general conditions for a fine covering family to be faithful for the Hausdorff dimension calculation and prove necessary and sufficient conditions for the Cantor series net-coverings to be faithful.
We start firstly with a very useful lemma, which can be proven easily, and, nevertheless, presents general necessary and sufficient conditions for the faithfulness.
Let us mention that this lemma can be obviously generalized to a multidimensional Euclidean space and even to any metric space, which can be equipped by fine covering families.
Based on the latter lemma one can easily get the following sufficient condition for the faithfulness of a fine covering family. 2) for any \delta > 0 there exists \varepsi 1 (\delta ) > 0 such that Then the family \Phi is faithful on [0, 1].
Let \{ n k \} \infty k=1 be a sequence with n k \in \BbbN \setminu \{ 1\} , k \in \BbbN . Let us recall that the expression of x in the following form which is said to be the Cantor covering family.
For a given m \in \BbbN let us consider 2 m probability measures \mu j , j = 0, 2 m -1 corresponding to the random variables , whose independent digits \xi j k have the following distributions: if k \not = 2 s , then \xi j k takes values 0, 1, ..., 2 k -1 with probabilities 1 2 k ; if k = 2 s , s \not = m, then \xi j k takes values 0, 1, ..., k \cdot 2 k -1 with probabilities 1 k\cdot 2 k ; if k = 2 m , then \xi j k takes values j \cdot 2 k + 0, j \cdot 2 k + 1, ..., (j + 1) \cdot 2 k -1 with probabilities 1 2 k . Let S j be the spectrum of the measure \mu j . From the construction of these measures and the definition of the set T it follows that S j \bigcap S i = \emptyse and T = \bigcup 2 m i=1 S i . Taking into account inequality \mathrm{ \mathrm{ \mu j (\Delta n(x)) \mathrm{ \mathrm{ \lambda (\Delta n(x)) \geq 1 2 , \forall x \in S j , and applying the mass distribution principle simultaneously for all measures m j , we get H Therefore, H 1 2 (T ) = 0. \square By using the same techniques it is not hard to prove the following result.
Proof. If \alpha = p q \in (0, 1) is a rational number, then the proof is completely similar to those in example 2, but in the definition of the set V k of digits which are admissible at the k-th step of construction of T \alpha , we define V k to be \{ 0, 1, .. If \alpha is an irrational number from (0, 1), then we choose an increasing sequence \{ p k q k \} of rational numbers converging to \alpha and apply the same technics with the following definition of the set V k := \{ 0, 1, ..., \Bigl[ (n k ) p k q k \Bigr] -1\} . \square Remark 3.3. The sequence n k has been chosen to be \{ 4 k \} only for the simplicity of calculations in the above examples. In the forthcoming paper we shall show how the latter statement can be generalized and give necessary and sufficient conditions for the Cantor net-coverings to be comparable.
Remark 3.4. The latter proposition shows extreme differences between comparable and faithful net-coverings and demonstrates that the class of faithful net-coverings is essentially wider then the class of comparable ones. The relation between these two classes is similar to the relation between bi-Lipshitz transformations and transformations preserving the Hausdorff dimension (see, e.g., [7,9] for details). More deep connections between faithfulness of net-coverings and the theory of transformations preserving the Hausdorff dimension will also be discussed in the forthcoming paper.