Hausdorff measures and the Hausdorff dimension are important tools in the study of fractals and singularly continuous probability measures. The determination or even estimation of the Hausdorff dimension of a set or measure is the crucial problem in fractal analysis, and a lot of research papers were devoted to these problems. Because of this reason, many interesting methods for the simplification of the procedure of the determination of the Hausdorff dimension were invented and developed during the last 20 years. One approach to such a simplification consists in some restrictions of admissible coverings. This idea came from Besicovitch’s works and has been used by Rogers and Taylor to construct comparable net measures [18] as approximations of the Hausdorff measures. In this paper, we develop this approach via construction of net coverings that lead to a special family of net measures, which are more general that comparable ones. We discuss the notion of faithfulness and nonfaithfulness of the family of cylinders generated by different systems of numerations for the Hausdorff dimension calculation.

Let us shortly recall that the α-dimensional Hausdorff measure of a set E⊂[0,1] with respect to a given fine family of coverings Φ is defined by Hα(E,Φ)=limϵ→0inf|Ej|≤ϵ∑j|Ej|α=limϵ→0Hϵα(E,Φ), where the infimum is taken over all at most countable ϵ-coverings {Ej} of E, Ej∈Φ . The nonnegative number dimH(E,Φ)=inf{α:Hα(E,Φ)=0} is called the Hausdorff dimension of the set E⊂[0,1] w.r.t. the family Φ. If Φ is the family of all subsets of [0,1] or Φ coincides with the family of all closed (open) subintervals of [0,1], then dimH(E,Φ) is equal to the classical Hausdorff dimension dimH(E) of a subset E⊂[0,1] .

A fine covering family Φ is said to be a faithful family of coverings (nonfaithful family of coverings) for the Hausdorff dimension calculation on [0,1] if dimH(E,Φ)=dimH(E),∀E⊆[0,1](resp.∃E⊆[0,1]:dimH(E,Φ)≠dimH(E)).

It is clear that any family Φ of comparable net-coverings (i.e., net-coverings that generate comparable net-measures) is faithful. Conditions for Vitali coverings to be faithful were studied by many authors (see, e.g., [2, 6, 7, 10] and the references therein). First steps in this direction have been done by Besicovitch [9], who proved the faithfulness of the family of cylinders of a binary expansion. His result was extended by Billingsley [10] to the family of s-adic cylinders, by Turbin and Pratsiovytyi [21] to the family of Q-S-cylinders, and by Albeverio and Torbin [2] to the family of Q∗ -cylinders for the matrices Q∗ with elements p0k,p(s−1)k bounded away from zero.

In all these papers, their authors used essentially the same approach to prove the faithfulness of the corresponding family of coverings: it has been shown that there exist positive constants C and n0∈N such that, for any ε>0 0$]]> and for any interval (a;b) with b−a<ε , there exist at most n0 cylinders from fine covering families that cover the interval (a,b) and their lengths do not exceed the value C(b−a) . It is rather obvious that such families Φ of cylinders generates comparable Hausdorff measures [18], and, therefore, they are faithful for the Hausdorff dimension calculation. Albeverio et al. [7] correctly mentioned that it was rather paradoxical that initial examples of nonfaithful families of coverings first appeared in the two-dimensional case (as a result of active studies of self-affine sets during the last decade of XX century, see, e.g., [8]). The family of cylinders of the classical continued fraction expansion can probably be considered as the first (and rather unexpected) example of nonfaithful one-dimensional net-family of coverings [16]. By using approach which has been invented by Yuval Peres to prove the nonfaithfulness of the family of continued fraction cylinders, in [7] the nonfaithfulness of the family Φ(Q∞) of cylinders of the Q∞ -expansion with polynomially decreasing elements  {qi} has been proven. This shows, in particular, that the family of cylinders of the classical Lüroth expansion is nonfaithful. Rather general sufficient conditions for Φ(Q∞) to be faithful were also obtained in [7, 15].

In 2012, Ibragim and Torbin [13] developed a new method to prove the faithfulness of the family of cylinders of Q∗ -expansion for the matrices Q∗ with elements p0k,p(s−1)k not tending to zero “too quickly.” In particular, they proved the following result.

This theorem, a generalization of [2], extended the family of faithful coverings generated by cylinders of Q∗ -expansion. In particular, we can easily apply this theorem to prove the faithfulness of the family of cylinders generated by the matrix Q∗=110…110k…12−110…12−110k…12−110…12−110k…110…110k….

On the other hand, if p0k,p(s−1)k tend to zero “too quickly” (e.g., s=4 , q0k=q3k=110k , q1k=q2k=12−110k ), then the above theorem does not work.

In the next section, we develop a new approach to prove the faithfulness of families of coverings and prove essentially new sufficient conditions for Q∗ -cylinders to be faithful (we do not need any information about the boundedness from zero of the elements q0k and q(s−1)k or any information about the rate of their convergence to zero).

From this corollary it follows, in particular, that the family Φ(Q∗) of cylinders generated by the matrix Q∗=110…110k…12−110…12−110k…12−110…12−110k…110…110k… is faithful.

Let us show how sufficient conditions for the faithfulness obtained in [2] can be easily derived from our results.

From the proof of the corollary it follows that it is easy to extend the results of [2] in the following way.

So, for the faithfulness of the family Φ(Q∗) , it suffices to control only elements of the first raw of the matrix Q∗ .

Let us also mention that Theorem 2.1 can give a positive answer on the faithfulness of Φ(Q∗) even for the case where infk{q0k,q(s−1)k}=0 and supikqik=1 simultaneously. To illustrate this, let us consider the matrix Q∗=1413…12n+113…1213…2n−12n13…1413…12n+113…, that is, q0k=q1k=q2k=13,k=2n,n∈N, and q0k=q2k=12n+1,q1k=2n−12n,k=2n−1,n∈N.

In such a case, infk{q0k,q(s−1)k}=0 and supikqik=1 , but it is clear that qkqk+1<13 for all k∈N , and, therefore, S(m,δ):= ∑k=1∞ ∏i=m+1m+kqiδ≤qm+1δ+2 ∑n=1∞(13)nδ=1+23δ−1,∀m∈N.

So, S(δ)<+∞ for all δ>0 0$]]>, which proves the faithfulness of the family  Φ(Q∗) .

Let {ξk} be a sequence of independent random variables taking values 0,1,…,s−1 with probabilities p0k,p1k,…,ps−1k , respectively. The random variable (4) ξ=Δξ1ξ2…ξk…Q∗ is said to be the random variable with independent Q∗ -symbols. Let νξ be the corresponding probability measure.

The Lebesgue structure of νξ is well studied (see, e.g., [2, 4]). It is known, in particular, that the distribution of ξ is of pure type. It is of pure discrete type if and only if (5) ∏k=1∞maxipik>0, 0,\]]]> of pure absolutely continuous type if and only if (6) ∏k=1∞p0kq0k+p1kq1k+⋯+p(s−1)kq(s−1)k>0, 0,\]]]> and of pure singularly continuous type if and only if infinite products (5) and (6) are equal to zero.

Let us recall that the Hausdorff dimension of the distribution of a random variable τ is defined as follows: dimH(τ)=inf{dimH(E),E∈Bτ}, where Bτ is the family of all possible (not necessarily closed) supports of the random variable τ, that is, Bτ={E:E∈B,Pτ(E)=1}.

Let us also recall the notion of the Hausdorff–Billingsley dimension of a set w.r.t. a probability measure and w.r.t. a system of partitions. Let υ be a continuous probability measure on [0,1] , and let Φ be the family of cylinders generated by some expansion. Then the (υ−α) -Hausdorff measure of a set E⊂[0,1] w.r.t. the family Φ and measure υ is defined as follows: Hα(E,υ,Φ)=limε→0[infυ(Ej)≤ε{∑jυα(Ej)}]=limε→0Hεα(E,υ,Φ), where Ej∈Φ,⋃jEj⊃E .

The number dimυ(E,Φ)=inf{α:Hα(E,υ,Φ)=0} is called the Hausdorff–Billingsley dimension of a set E w.r.t. υ and Φ.

Let hj:=− ∑i=0s−1pijlnpij,0ln0:=0,Hn:= ∑j=1nhj,bj:=− ∑i=0s−1pijlnqij,Bn:= ∑j=1nbj,