1 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, 12–15]). All these measures are the distributions of random variables of the form
where $\{\xi _{k}\}$ 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 $Q_{\infty }$-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 $\mu _{\xi }$ 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.
2 On GLS-expansion and fractal properties of related probability measures
Let $Q_{\infty }=(q_{0},q_{1},\dots ,q_{n},\dots \hspace{0.1667em})$ be an infinite stochastic vector with positive coordinates. Let us consider a countable sequence $\varDelta _{i}=[a_{i},b_{i}]$ of intervals such that $\mathit{Int}(\varDelta _{i})\cap \mathit{Int}(\varDelta _{j})=\varnothing \hspace{2.5pt}(i\ne j)$ and $|\varDelta _{i}|=q_{i}$. The sets $\varDelta _{i}$ 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 $\varDelta _{i_{1}}$ of rank 1, we consider a sequence of nonoverlapping closed intervals $\varDelta _{i_{1}i_{2}}\subset \varDelta _{i_{1}}$ such that
and the placement of $\varDelta _{i_{1}i_{2}}$ in $\varDelta _{i_{1}}$ is the same as $\varDelta _{i_{1}}$ in $[0;1]$. The closed intervals $\varDelta _{i_{1}i_{2}}$ are said to be cylinders of rank 2 of the GLS-expansion.
Similarly, for every cylinder of rank $(n-1)$ $\varDelta _{i_{1}i_{2}\dots i_{n-1}}$, we consider the sequence of nonoverlapping closed intervals $\varDelta _{i_{1}i_{2}\dots i_{n}}\subset \varDelta _{i_{1}i_{2}\dots i_{n-1}}$ such that
\[\frac{|\varDelta _{i_{1}i_{2}\dots i_{n}}|}{|\varDelta _{i_{1}i_{2}\dots i_{n-1}}|}=q_{i_{n}},\hspace{1em}i\in N\cup \{0\},\]
and the placement of $\varDelta _{i_{1}i_{2}\dots i_{n}}$ in $\varDelta _{i_{1}i_{2}\dots i_{n-1}}$ is the same as $\varDelta _{i_{1}}$ in $[0;1]$.The closed intervals $\varDelta _{i_{1}i_{2}\dots i_{n}}$ are said to be cylinders of rank n of the GLS-expansion. From the construction it follows that
\[|\varDelta _{i_{1}i_{2}...i_{n}}|=q_{i_{1}}\cdot q_{i_{2}}\cdot \dots \cdot q_{i_{n}}\le {(q_{\max })}^{n}\to 0\hspace{1em}(n\to \infty ),\]
where $q_{\max }:=\max _{i}q_{i}$.So, for any sequence of indices $\{i_{k}\}$ ($i_{k}\in N\cup \{0\})$, there exists the sequence of embedded closed intervals
\[\varDelta _{i_{1}}\subset \varDelta _{i_{1}i_{2}}\subset \varDelta _{i_{1}i_{2}i_{2}}\subset \cdots \subset \varDelta _{i_{1}i_{2}\dots i_{k}}\subset \cdots \]
with $|\varDelta _{i_{1}i_{2}\dots i_{k}}|\to 0,\hspace{2.5pt}k\to \infty $. Therefore, there exists a unique point $x\in [0,1]$ that belongs to all these cylinders.Conversely, if $x\in [0,1]$ belongs to some cylinder of rank k for any $k\in N$ and x is not an end-point for any cylinder, then there exists a unique sequence of the cylinders
\[\varDelta _{i_{1}(x)}\supset \varDelta _{i_{1}(x)i_{2}(x)}\supset \varDelta _{i_{1}(x)i_{2}(x)i_{3}(x)}\supset \cdots \supset \varDelta _{i_{1}(x)i_{2}(x)\dots i_{k}(x)}\supset \cdots \]
containing x, and
\[x=\bigcap \limits_{k=1}^{\infty }\varDelta _{i_{1}(x)i_{2}(x)\dots i_{k}(x)}=\varDelta _{i_{1}(x)i_{2}(x)\dots i_{k}(x)\dots }.\]
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 $Q_{\infty }$-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 $\varphi =\frac{1+\sqrt{5}}{2}$, we get the ${G_{\infty }^{2}}$-expansion of x [11].
Let $Q_{\infty }=(q_{0},q_{1},\dots ,q_{n},\dots \hspace{0.1667em})$ be a stochastic vector with positive coordinates, and let $x={\varDelta _{i_{1}(x)i_{2}(x)\dots i_{k}(x)\dots }^{\mathit{GLS}}}$ be the GLS-expansion of $x\in [0,1]$.
Let $\{\xi _{k}\}$ be a sequence of independent identically distributed random variables:
where
Using the sequence $\{\xi _{k}\}$ and a given GLS-expansion, let us consider the random variable
which is said to be the random variable with independent identically distributed GLS-symbols. Let $\mu _{\xi }$ 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 $N_{0}:=\{0,1,2,\dots \hspace{0.1667em}\}$, and let
If the set V is finite, then $C[\mathit{GLS},V]$ 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 $Q_{\infty }$ and subsets V such that equation (1) has no roots on the unit interval.
For example, if $q_{i}=\frac{A}{(i+2){\ln }^{2}(i+2)}$ and $V=N$, then the equation $\sum _{i\in V}{q_{i}^{x}}=1$ has no roots on $[0;1]$.
Proof.
First, let us show that for any $k\in N$, the set $C[\mathit{GLS},V]$ can be covered by cylinders of rank k and that the $\alpha _{0}$-volume of this covering is equal to 1.
For $k=1$, the set $C[\mathit{GLS},V]$ can be covered by cylinders of rank 1. It easy to see that the $\alpha _{0}$-volume is equal to 1:
Suppose that for $k=n-1$, the $\alpha _{0}$-volume of the covering of $C[\mathit{GLS},V]$ by cylinders of rank $n-1$ is equal to 1. Let us show that for $k=n$, the $\alpha _{0}$-volume of the covering of $C[\mathit{GLS},V]$ by cylinders of rank n will not change. We have
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \sum \limits_{i_{j}\in V}|\varDelta _{i_{1}i_{2}...i_{n-1}i_{n}}{|}^{\alpha _{0}}& \displaystyle =\sum \limits_{i_{j}\in V}{(q_{i_{1}}q_{i_{2}}\dots q_{n-1}q_{n})}^{\alpha _{0}}\\{} & \displaystyle =\sum \limits_{i_{1}\in V}{q_{i_{1}}^{\alpha _{0}}}\cdot \sum \limits_{i_{j}\in V}{(q_{i_{1}}q_{i_{2}}\dots q_{n-1})}^{\alpha _{0}}=1.\end{array}\]
So, for any $\varepsilon >0$, we get
Hence,
By the definition of the Hausdorff–Besicovitch dimension we get
Let us show that $\dim _{H}(C[\mathit{GLS},V])\ge \alpha _{0}$. To this end, let us consider sets $V=\{i_{1},\dots ,i_{k},\dots \hspace{0.1667em}\}$, $V_{k}=\{i_{1},\dots ,i_{k}\},\hspace{2.5pt}k\ge 2,\hspace{2.5pt}k\in N$, and the sequence $C[\mathit{GLS},V_{k}]$ of subsets of $C[\mathit{GLS},V]$. For all $k\ge 2,\hspace{2.5pt}k\in N$, we have
and, therefore,
Let $\dim _{H}(C[\mathit{GLS},V_{k}])=\alpha _{k}$. The sets $C[\mathit{GLS},V_{k}]$ are self-similar and satisfy the open set condition (OSC). Hence, the Hausdorff–Besicovitch dimension $\alpha _{k}$ of $C[\mathit{GLS},V_{k}]$ coincides with the solution of the equation
It is clear that $\alpha _{2}<\alpha _{3}<\cdots <\alpha _{k}<\cdots \hspace{0.1667em}$ and $\alpha _{k}<\alpha _{0}$. So, the sequence $\{\alpha _{k}\}$ is increasing and bounded. Therefore, there exists a limit $\lim _{k\to \infty }\alpha _{k}={\alpha }^{\ast }$.
It is clear that ${\alpha }^{\ast }\le \alpha _{0}$ because $\alpha _{k}<\alpha _{0}\hspace{2.5pt}(\forall k\in N)$. Let us prove that ${\alpha }^{\ast }=\alpha _{0}$.
Assume the opposite: let ${\alpha }^{\ast }<\alpha _{0}$. Then there exists ${\alpha ^{\prime }}$ such that ${\alpha }^{\ast }<{\alpha ^{\prime }}<\alpha _{0}$. Then $\sum _{i\in V_{k}}{q_{i}^{{\alpha ^{\prime }}}}<1$ for all $k\in N$. Since $\sum _{i\in V_{k}}{q_{i}^{\alpha _{k}}}=1$, we get $\sum _{i\in V_{k}}{q_{i}^{{\alpha ^{\prime }}}}<1$ for all $k\in N$. Let us consider the series ${\sum _{k=1}^{\infty }}{q_{i_{k}}^{{\alpha ^{\prime }}}}$. It is clear that ${\sum _{k=1}^{n}}{q_{i_{k}}^{{\alpha ^{\prime }}}}<1$ for all $n\in N$. So $\lim _{n\to \infty }{\sum _{k=1}^{n}}{q_{i_{k}}^{{\alpha ^{\prime }}}}\le 1$. Therefore, $\sum _{i\in V}{q_{i}^{{\alpha ^{\prime }}}}\le 1$.
Since ${q_{i}^{{\alpha ^{\prime }}}}>{q_{i}^{\alpha _{0}}}$ for all $i\in V$ and $\sum _{i\in V}{q_{i}^{\alpha _{0}}}=1$, we get $\sum _{i\in V}{q_{i}^{{\alpha ^{\prime }}}}>\sum _{i\in V}{q_{i}^{\alpha _{0}}}=1$, which contradicts the already proven inequality $\sum _{i\in V}{q_{i}^{{\alpha ^{\prime }}}}\le 1$. This proves that ${\alpha }^{\ast }=\alpha _{0}$.
Theorem 2.
If the matrix $Q_{\infty }$ and the set $V=\{i_{1},i_{2},\dots ,i_{k},i_{k+1},\dots \hspace{0.1667em}\}$ are such that equation $\sum _{i\in V}{q_{i}^{x}}=1$ has no roots on $[0,1]$, then
\[\dim _{H}\big(C[\mathit{GLS},V]\big)=\underset{k\to \infty }{\lim }\dim _{H}\big([\mathit{GLS},V_{k}]\big),\]
where $V_{k}=\{i_{1},i_{2},\dots ,i_{k}\},\hspace{2.5pt}k\in N,\hspace{2.5pt}k\ge 2$.
Proof.
The sets $C[\mathit{GLS},V_{k}]$ are self-similar and satisfy the OSC. Thus the dimension $\alpha _{k}$ can be obtained as a solution of the equation $\sum _{i\in V_{k}}{q_{i}^{x}}=1$. It is easy to see that
Therefore, there exists the limit
It is clear that ${\alpha }^{\ast }\le \dim _{H}(C[\mathit{GLS},V])$ because $C[\mathit{GLS},V_{k}]\subset C[\mathit{GLS},V]$ for all $k\ge 2$. Suppose that ${\alpha }^{\ast }<\dim _{H}(C[\mathit{GLS},V])$. Then there exists a number ${\alpha ^{\prime }}$ such that ${\alpha }^{\ast }<{\alpha ^{\prime }}<\dim _{H}(C[\mathit{GLS},V])$. It is clear that
So, $\sum _{i\in V}{q_{i}^{{\alpha ^{\prime }}}}\le 1$.
On the other hand, ${H}^{{\alpha ^{\prime }}}(C[\mathit{GLS},V])=\infty $ by the definition of the Hausdorff–Besicovitch dimension (because ${\alpha ^{\prime }}<\dim _{H}(C[\mathit{GLS},V])$). Then the ${\alpha ^{\prime }}$-dimensional Hausdorff measure of the set $C[\mathit{GLS},V]$ with respect to the family Φ of coverings that are generated by the GLS-expansion of the unit segment is equal to ${H}^{{\alpha ^{\prime }}}(C[\mathit{GLS},\varPhi ])=\infty $ (where the family Φ is a locally fine system of the coverings of the unit segment, i.e., for any $\varepsilon >0$, there exists such a covering of $[0,1]$ by the subsets $E_{j}\in \varPhi $ such that $|E_{j}|<\varepsilon $ and $[0,1]=\bigcup _{j}E_{j}$). Since the set $C[\mathit{GLS},V]$ 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
\[\begin{array}{r@{\hskip0pt}l}\displaystyle \sum \limits_{i_{q}\in V,q\in \{1,\dots ,k\}}|\varDelta _{i_{1}i_{2}\dots i_{k}}{|}^{{\alpha ^{\prime }}}& \displaystyle >M,\\{} \displaystyle \sum \limits_{i_{q}\in V,q\in \{1,\dots ,k\}}|\varDelta _{i_{1}i_{2}\dots i_{k}}{|}^{{\alpha ^{\prime }}}& \displaystyle =\sum \limits_{i_{q}\in V,q\in \{1,\dots ,k-1\}}|\varDelta _{i_{1}i_{2}\dots i_{k-1}}{|}^{{\alpha ^{\prime }}}\cdot \sum \limits_{i_{k}\in V}{\varDelta _{i_{k}}^{{\alpha ^{\prime }}}}\\{} & \displaystyle <\sum \limits_{i_{q}\in V,q\in \{1,\dots ,k-1\}}|\varDelta _{i_{1}i_{2}\dots i_{k-1}}{|}^{{\alpha ^{\prime }}}\\{} & \displaystyle =\sum \limits_{i_{q}\in V,q\in \{1,\dots ,k-2\}}|\varDelta _{i_{1}i_{2}\dots i_{k-2}}{|}^{{\alpha ^{\prime }}}\cdot \sum \limits_{i_{k-1}\in V}{q_{i_{k-1}}^{{\alpha ^{\prime }}}}\\{} & \displaystyle <\sum \limits_{i_{q}\in V,q\in \{1,\dots ,k-2\}}|\varDelta _{i_{1}i_{2}\dots i_{k-2}}{|}^{{\alpha ^{\prime }}}<\cdots \\{} & \displaystyle <\sum \limits_{i_{1}\in V}|\varDelta _{i_{1}}{|}^{{\alpha ^{\prime }}}=\sum \limits_{i_{1}\in V}{q_{i_{1}}^{{\alpha ^{\prime }}}}<1.\end{array}\]
Proof.
Let $\alpha _{0}=\dim _{H}(C[\mathit{GLS},V])$. Show that $\sup \{x:\sum _{i\in V}{q_{i}^{x}}\ge 1\}\ge \alpha _{0}$. Let us consider the function
and denote the set
Let $\alpha _{k}$ and $\alpha _{k+1}$ be the solutions of the equations
and
respectively.
Let us show that $\alpha _{k}<\alpha _{k+1}<\alpha _{0}$. If $\alpha _{0}$ is the solution of $\sum _{i\in V}{q_{i}^{x}}=1$, then it is easy to see that $\alpha _{k}<\alpha _{k+1}<\alpha _{0}$. If $\sum _{i\in V}{q_{i}^{x}}=1$ has no roots on $[0,1]$, then
and $\alpha _{k}<\alpha _{k+1}$, so that $\alpha _{k}<\alpha _{k+1}<\alpha _{0}$.
Express the function $\varphi (x)$ as follows:
\[\varphi (x)=\underset{\ge 1}{\underbrace{{q_{i_{1}}^{x}}+\cdots +{q_{i_{k}}^{x}}+{q_{i_{k+1}}^{x}}}}+\sum \limits_{j=k+1}^{\infty }{q_{i_{j}}^{x}}.\]
It is easy to see that for all $x\in [\alpha _{k},\alpha _{k+1}]$ $(x<\alpha _{0},\hspace{2.5pt}k\in N)$, $\varphi (x)\ge 1$. Then $A_{+}\supset (-\infty ;\alpha _{0})$ and $\sup A_{+}\ge \alpha _{0}$.Let us show that $\sup A_{+}\le \alpha _{0}$. Suppose the opposite. If $\sup A_{+}>\alpha _{0}$, then there exists $x_{1}$ such that $x_{1}\in (\alpha _{0};\sup A_{+}]$, $x_{1}\in A_{+}$, and $\varphi (x_{1})\ge 1$. So
Since $\alpha _{k}$ is a solution of $\sum _{i\in V_{k}}{q_{i}^{x}}=1$ and $\alpha _{k}<\alpha _{0}$, we get
and
It is clear that
and
So, from the obtained contradiction it follows that $\sup A_{+}=\alpha _{0}$. □
Let ${\varDelta _{\infty }^{\mathit{GLS}}}$ be the set of those $x\in [0;1]$ that do not belong to any cylinder of the first rank of the GLS-expansion. The set ${\varDelta _{\infty }^{\mathit{GLS}}}$ 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 $E_{j}\hspace{2.5pt}(\dim _{H}(E_{i}\cap E_{j})<\dim _{H}E,i\ne j)$ such that the set E is similar to the sets $E_{j}$ with coefficient $k_{j}$.
Since the spectrum $S_{\xi }$ of the distribution of a random variable ξ with independent identically distributed GLS-symbols is a self-similar or N-self-similar set and $S_{\xi }={(C[\mathit{GLS},V])}^{\mathit{cl}}$, we can apply the above results to calculate the Hausdorff–Besicovitch dimension of the spectrum $S_{\xi }$ for the case where ${\varDelta _{\infty }^{\mathit{GLS}}}$ is an at most countable set.
Theorem 4.
Let $V:=\{i:p_{i}>0\}$. If ${\varDelta _{\infty }^{\mathit{GLS}}}$ is at most countable, then the Hausdorff–Besicovitch dimension of the spectrum of the distribution of a random variable ξ with independent identically distributed $\mathit{GLS}$-symbols can be calculated in the following way.
1) If the equation $\sum _{i\in V}{q_{i}^{x}}=1$ has one root $\alpha _{0}$ on $[0,1]$, then
