On spectra of probability measures generated by GLS-expansions

We study properties of distributions of random variables with independent identically distributed symbols of generalized L\"{u}roth series (GLS) expansions (the family of GLS-expansions contains L\"{u}roth expansion and $Q_{\infty}$- and $G_{\infty}^2$-expansions). To this end, we explore fractal properties of the family of Cantor-like sets $C[\mathit{GLS},V]$ consisting of real numbers whose GLS-expansions contain only symbols from some countable set $V\subset N\cup\{0\}$, and derive exact formulae for the determination of the Hausdorff--Besicovitch dimension of $C[\mathit{GLS},V]$. Based on these results, we get general formulae for the Hausdorff--Besicovitch dimension of the spectra of random variables with independent identically distributed GLS-symbols for the case where all but countably many points from the unit interval belong to the basis cylinders of GLS-expansions.


Introduction
During the last 20 years, many authors studied singularly continuous probability measures generated by different expansions of real numbers (see, e.g., [2,9,10,[12][13][14][15]). All these measures are the distributions of random variables of the form where {ξ k } are independent or Markovian, and F stands for some expansion of real numbers. For the case of expansions over finite alphabets, fractal properties of the spectra of the corresponding measures are relatively well studied. For the case of infinite alphabets, the situation is essentially more complicated. In [8] and [9], it has been shown that even for self-similar Q ∞ -expansion and for i.i.d. case, the Hausdorff-Besicovitch dimension of the corresponding spectra cannot be calculated in a traditional way (as a root of the corresponding equation), and formulae for the Hausdorff dimension of the measure µ ξ are also unknown.
In this paper, we generalize results from [8] and [9] for the case of distributions of random variables with independent identically distributed GLS digits and get general formulae for the determination of the Hausdorff-Besicovitch dimension of spectra of ξ for the case where all but countably many points from the unit interval belong to the basis cylinders of GLS-expansion.

On GLS-expansion and fractal properties of related probability measures
Let Q ∞ = (q 0 , q 1 , . . . , q n , . . . ) be an infinite stochastic vector with positive coordinates. Let us consider a countable sequence The sets ∆ i are said to be cylinders of GLS-expansion (generalized Lüroth series).
Let us remark that the placement of cylinders of rank 1 is completely determined by the preselected procedure.
For every cylinder ∆ i1 of rank 1, we consider a sequence of nonoverlapping closed intervals ∆ i1i2 ⊂ ∆ i1 such that and the placement of ∆ i1i2 in ∆ i1 is the same as ∆ i1 in [0; 1]. The closed intervals ∆ i1i2 are said to be cylinders of rank 2 of the GLS-expansion. Similarly, for every cylinder of rank (n−1) ∆ i1i2...in−1 , we consider the sequence of nonoverlapping closed intervals ∆ i1i2...in ⊂ ∆ i1i2...in−1 such that The closed intervals ∆ i1i2...in are said to be cylinders of rank n of the GLSexpansion. From the construction it follows that where q max := max i q i .
So, for any sequence of indices {i k } (i k ∈ N ∪ {0}), there exists the sequence of embedded closed intervals with |∆ i1i2...i k | → 0, k → ∞. Therefore, there exists a unique point x ∈ [0, 1] that belongs to all these cylinders.
Conversely, if x ∈ [0, 1] belongs to some cylinder of rank k for any k ∈ N and x is not an end-point for any cylinder, then there exists a unique sequence of the cylinders The latter expression is called the GLS-expansion of x (see, e.g., [1,3,4,6,7] for details).
Let us remark that the Lüroth expansion and Q ∞ -expansion [8,9] are particular cases of the GLS-expansion. For the case where the ratio of lengths of two embedded cylinders of successive ranks depends on the last index and it is a power of ϕ = 1+ . Let Q ∞ = (q 0 , q 1 , . . . , q n , . . . ) be a stochastic vector with positive coordinates, and let . Let {ξ k } be a sequence of independent identically distributed random variables: Using the sequence {ξ k } and a given GLS-expansion, let us consider the random variable ξ = ∆ GLS ξ1ξ2...ξ k ... , which is said to be the random variable with independent identically distributed GLSsymbols. Let µ ξ be the corresponding probability measure.
To investigate metric, topological, and fractal properties of the spectrum of the random variable with independent identically distributed GLS-symbols, let us study properties of the following family of sets. Let V be a subset of is a self-similar set satisfying the open set condition (see, e.g., [5]). So, its Hausdorff-Besicovitch dimension coincides with the root of the equation If the set V is countable, then the situation is essentially more complicated. In particular, there exist stochastic vectors Q ∞ and subsets V such that equation (1) has no roots on the unit interval.
For example, if q i = A (i+2) ln 2 (i+2) and V = N , then the equation i∈V q x i = 1 has no roots on [0; 1]. Theorem 1. If a stochastic vector Q ∞ and a set V ⊂ N 0 are such that the equation Proof. First, let us show that for any k ∈ N , the set C[GLS , V ] can be covered by cylinders of rank k and that the α 0 -volume of this covering is equal to 1. For k = 1, the set C[GLS , V ] can be covered by cylinders of rank 1. It easy to see that the α 0 -volume is equal to 1: Suppose that for k = n − 1, the α 0 -volume of the covering of C[GLS , V ] by cylinders of rank n − 1 is equal to 1. Let us show that for k = n, the α 0 -volume of the covering of C[GLS , V ] by cylinders of rank n will not change. We have So, for any ε > 0, we get and, therefore, It is clear that α 2 < α 3 < · · · < α k < · · · and α k < α 0 . So, the sequence {α k } is increasing and bounded. Therefore, there exists a limit lim k→∞ α k = α * . It is clear that α * ≤ α 0 because α k < α 0 (∀k ∈ N ). Let us prove that α * = α 0 . Assume the opposite: let α * < α 0 . Then there exists α ′ such that α * < α ′ < α 0 . Then i∈V k q α ′ i < 1 for all k ∈ N . Since i∈V k q α k i = 1, we get i∈V k q α ′ i < 1 for all k ∈ N . Let us consider the series i for all i ∈ V and i∈V q α0 i = 1, we get i∈V q α ′ i > i∈V q α0 i = 1, which contradicts the already proven inequality i∈V q α ′ i ≤ 1. This proves that α * = α 0 .
Since for any k ≥ 2, k ∈ N , Thus, Proof. The sets C[GLS , V k ] are self-similar and satisfy the OSC. Thus the dimension α k can be obtained as a solution of the equation i∈V k q x i = 1. It is easy to see that Therefore, there exists the limit Φ]) = ∞ (where the family Φ is a locally fine system of the coverings of the unit segment, i.e., for any ε > 0, there exists such a covering of [0, 1] by the subsets E j ∈ Φ such that |E j | < ε and [0, 1] = j E j ). Since the set C[GLS , V ] can be covered by cylindrical segments of the GLS-expansion with indices from V , we deduce that for any M > 0, there exists k(M ) such that for all k > k(M ), we have the inequality iq ∈V,q∈{1,...,k} From the obtained contradiction it follows that Remark 1. Theorems 1 and 2 can be considered as natural generalizations of results from [8].
It is clear that i∈V q α0 i < 1 and i∈V q x1 i ≤ 1.
So, from the obtained contradiction it follows that sup A + = α 0 .
Let ∆ GLS ∞ be the set of those x ∈ [0; 1] that do not belong to any cylinder of the first rank of the GLS-expansion. The set ∆ GLS ∞ can be empty, countable, or of continuum cardinality.
Let us recall that the nonempty and bounded set E is called N -self-similar if it can be represented as a union of a countably many sets E j (dim H (E i ∩ E j ) < dim H E, i = j) such that the set E is similar to the sets E j with coefficient k j .
Since the spectrum S ξ of the distribution of a random variable ξ with independent identically distributed GLS-symbols is a self-similar or N -self-similar set and S ξ = (C[GLS , V ]) cl , we can apply the above results to calculate the Hausdorff-Besicovitch dimension of the spectrum S ξ for the case where ∆ GLS ∞ is an at most countable set. So, we get the following theorem, which can be considered as a corollary of Theorems 1 and 2.