1 Introduction
Singularly continuous probability measures were studied during almost all XX century and there are a lot of open problems related to them. The fractal and multifractal approaches to the study of such measures are known to be extremely useful (see, e.g., [7, 12, 39] and references therein). The study of fractal properties of different families of singularly continuous probability measures (see, e.g., [7, 16, 26, 28, 27, 29, 31, 38, 1, 42] and references therein) can be used to solve non-trivial problems in the metric number theory ([8, 9, 5, 4, 10, 30, 21]), in the theory of dynamical systems and DP-transformations and in fractal analysis ([6, 3, 11, 19, 18, 17, 22, 41, 43]).
On the other hand, for many families of probability measures the problem “singularity vs absolute continuity” is extremely complicated even for the so-called probability distributions of the Jessen–Wintner type, i.e., distributions of random variables which are sums of almost surely convergent series of independent discretely distributed random variables (such probability distributions are of pure type [23]). The Lévy theorem [25] gives necessary and sufficient conditions for such measures to be discrete resp. continuous, and the main problem is to find sharp conditions for absolute resp. singular continuity. Infinite Bernoulli convolutions make an important subclass of such measures (see, e.g., [2, 32, 34, 36, 35, 37] and references therein). Another wide family of probability distributions where the problem “singularity vs absolute continuity” is still open consists of probability distributions of the following form:
where $\xi _{n}$ are independent symbols of some generalized F-expansion over some alphabet A. Random variables with independent symbols of s-adic expansions, continued fraction expansions, the Lüroth expansion, the Sylvester and Engel expansions are among them. This paper is devoted to the development of probabilistic theory of Oppenheim expansions of real numbers which contains many important expansions as rather special cases. Let us mention that many authors studied normal properties of real numbers in terms of digits of their Oppenheim expansion and the Hausdorff dimension of corresponding exceptional sets (see, e.g., [13, 15, 44, 45, 24]). In Section 2 we develop approach which has been invented by G. Torbin to study normal properties of the Ostrogradsky–Sierpinski–Pierce expansion [40] and get general results on normal properties of Oppenheim expansions. Based on these results in the last section of the paper we show that singularity is typical for the family of probability measures with independent symbols of ROE expansions.
2 On metric theory of the restricted Oppenheim expansion
It is known ([14]) that any real number $x\in (0,1)$ can be represented in the form of the Oppenheim expansion
where $a_{n}=a_{n}(d_{1},\dots ,d_{n})$, $b_{n}=b_{n}(d_{1},\dots ,d_{n})$ are positive integer valued functions and the denominators $d_{n}$ are determined by the following procedure: for a given x we define the sequences $\{x_{n}\}$ and $\{d_{n}\}$ via
(1)
\[ x\sim \frac{1}{d_{1}}+\frac{a_{1}}{b_{1}}\frac{1}{d_{2}}+\cdots +\frac{a_{1}a_{2}\cdot \dots \cdot a_{n}}{b_{1}b_{2}\cdot \dots \cdot b_{n}}\frac{1}{d_{n+1}}+\cdots \]A sufficient condition for a series on the right-hand side in (1) to be the expansion of its sum is:
We call the expansion (1) the restricted Oppenheim expansion (ROE) of x if $a_{n}$ and $b_{n}$ depend only on the last denominator $d_{n}$ and if the function
is integer valued.
Let us consider some examples of the restricted Oppenheim expansions.
Let us mention that metric, dimensional and probabilistic theories of Oppenheim series are not sufficiently developed. In fact, as evidenced by recent works and thesis in the field ([46, 20, 33]), even such partial cases of Oppenheim expansions as the Lüroth series, the Engel and Sylvester series generate a number of challenges for the metric and probabilistic number theory. The main purpose of this article is to develop some general methods of the metric theory of numbers and Oppenheim expansions and to show their effectiveness in the study of Lebesgue structures of distributions of random variables with independent symbols of Oppenheim expansions.
Choose the probability space $(\varOmega ,\mathcal{A},P)$, with $\varOmega =(0,1)$, $\mathcal{A}$ the set of Lebesgue measurable subsets of $(0,1)$ and the Lebesgue measure as P.
Let ${\bigtriangleup _{j_{1}j_{2}\dots j_{n}}^{\mathit{ROE}}}:=\{x:d_{1}(x)=j_{1},d_{2}(x)=j_{2},\dots ,d_{n}(x)=j_{n}\}$ be the cylinder of rank n with base $(j_{1},j_{2},\dots ,j_{n})$.
Lemma 1.
([14]) Let x be the random variable, which is uniformly distributed on the unit interval and let $d_{j}:=d_{j}(x)$. Then
\[ P(d_{1}=j_{1},\dots ,d_{n}=j_{n})=\frac{a_{1}a_{2}\cdot \dots \cdot a_{n-1}}{b_{1}b_{2}\cdot \dots \cdot b_{n-1}}\frac{1}{j_{n}(j_{n}-1)}\]
where $a_{i}=a_{i}(j_{i})$, $b_{i}=b_{i}(j_{i})$ $(i=1,2,\dots ,n-1)$.Therefore, we get the following properties of cylinders:
-
1) ${\bigtriangleup _{j_{1}j_{2}\dots j_{n-1}}^{\mathit{ROE}}}={\bigcup \limits_{i=1}^{\infty }}{\bigtriangleup _{j_{1}j_{2}\dots j_{n-1}i}^{\mathit{ROE}}}$.
-
2) $\mathit{sup}{\bigtriangleup _{j_{1}j_{2}\dots j_{n}}^{\mathit{ROE}}}=\mathit{inf}{\bigtriangleup _{j_{1}j_{2}\dots j_{n-1}(j_{n}-1)}^{\mathit{ROE}}}$.
-
3) $\mathit{inf}{\bigtriangleup _{j_{1}j_{2}\dots j_{n}}^{\mathit{ROE}}}=\frac{1}{j_{1}}+\frac{a_{1}}{b_{1}}\frac{1}{j_{2}}+\cdots +\frac{a_{1}a_{2}\dots a_{n}}{b_{1}b_{2}\dots b_{n}}\frac{1}{j_{n}}$,$\mathit{sup}{\bigtriangleup _{j_{1}j_{2}\dots j_{n}}^{\mathit{ROE}}}=\frac{1}{j_{1}}+\frac{a_{1}}{b_{1}}\frac{1}{j_{2}}+\cdots +\frac{a_{1}a_{2}\dots a_{n-1}}{b_{1}b_{2}\dots b_{n-1}}\frac{1}{j_{n}-1}$.
-
4) $|{\bigtriangleup _{j_{1}j_{2}\dots j_{n}}^{\mathit{ROE}}}|=\frac{a_{1}a_{2}\dots a_{n-1}}{b_{1}b_{2}\dots b_{n-1}}\frac{1}{j_{n}(j_{n}-1)}$.
If the first k symbols of ROE are fixed, then $(k+1)$-st symbol of ROE cannot take values $2,3,\dots ,\frac{a_{k}}{b_{k}}d_{k}(d_{k}-1)$, $\forall k\in \mathbb{N}$.
Each of the cylinders of ROE can be uniquely rewritten in terms of the difference restricted Oppenheim expansion ($\overline{\mathit{ROE}}$):
Then series (1) can be rewritten as follows:
\[ x=\frac{1}{\alpha _{1}+1}+\frac{a_{1}}{b_{1}}\frac{1}{\frac{a_{1}}{b_{1}}d_{1}(d_{1}-1)+\alpha _{2}}+\frac{a_{1}a_{2}}{b_{1}b_{2}}\frac{1}{\frac{a_{2}}{b_{2}}d_{2}(d_{2}-1)+\alpha _{3}}+\cdots =:{\varDelta _{\alpha _{1}\alpha _{2}\dots \alpha _{n}\dots }^{\overline{\mathit{ROE}}}}.\]
where $\alpha _{k}\in \{1,2,3,\dots \}$. Theorem 2.
If there exists a sequence $l_{k}$, such that $\forall x\in [0,1]$:
and series
then for any digit $i_{0}$ almost all (with respect to the Lebesgue measure) real numbers $x\in [0,1]$ contain symbol $i_{0}$ only finitely many times in $\overline{\mathit{ROE}}$.
Proof.
Let $N_{i}(x)$ be a number of symbols “i” in $\overline{\mathit{ROE}}$ of number x. Let us prove that the Lebesgue measure of the set $A_{i}=\{x:N_{i}(x)=\infty \}$ is equal to 0 for all $i\in \mathbb{N}$.
Consider the set
From the definition of the set ${\bar{\bigtriangleup }_{i}^{k}}$ and properties of cylindrical sets it follows that
Let us consider the following ratio:
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \frac{|{\bigtriangleup _{\alpha _{1}\dots \alpha _{k-1}i}^{\overline{\mathit{ROE}}}}|}{|{\bigtriangleup _{\alpha _{1}\dots \alpha _{k-1}}^{\overline{\mathit{ROE}}}}|}\\{} & \displaystyle =\frac{|{\bigtriangleup _{d_{1}\dots d_{k-1}(\frac{a_{k-1}}{b_{k-1}}d_{k-1}(d_{k-1}-1)+i)}^{\mathit{ROE}}}|}{|{\bigtriangleup _{d_{1}\dots d_{k-1}}^{\mathit{ROE}}}|}\\{} & \displaystyle =\frac{a_{1}\dots a_{k-1}}{b_{1}\dots b_{k-1}}\cdot \frac{1}{(\frac{a_{k-1}}{b_{k-1}}d_{k-1}(d_{k-1}-1)+i)(\frac{a_{k-1}}{b_{k-1}}d_{k-1}(d_{k-1}-1)+i-1)}\\{} & \displaystyle :\frac{a_{1}\dots a_{k-2}}{b_{1}\dots b_{k-2}}\cdot \frac{1}{d_{k-1}(d_{k-1}-1)}\\{} & \displaystyle \le \frac{a_{k-1}}{b_{k-1}}\cdot \frac{d_{k-1}(d_{k-1}-1)}{\frac{a_{k-1}}{b_{k-1}}d_{k-1}(d_{k-1}\hspace{0.1667em}-\hspace{0.1667em}1)}\cdot \frac{1}{\frac{a_{k-1}}{b_{k-1}}d_{k-1}(d_{k-1}\hspace{0.1667em}-\hspace{0.1667em}1)}=\frac{b_{k-1}}{a_{k-1}}\cdot \frac{1}{d_{k-1}(d_{k-1}\hspace{0.1667em}-\hspace{0.1667em}1)}<l_{k}\end{array}\]
Then
It is clear, that the set $A_{i}$ is the upper limit of the sequence of sets $\{{\bar{\bigtriangleup }_{i}^{k}}\}$, i.e.,
Example 4.
Consider the Sylvester series:
\[\begin{array}{r@{\hskip0pt}l}\displaystyle d_{1}& \displaystyle \in \{2,3,\dots \},\\{} \displaystyle d_{k+1}& \displaystyle =d_{k}(d_{k}-1)+i,\hspace{1em}i\in \{1,2,3,\dots \}.\end{array}\]
If $d_{1}=2$, then the minimal admissible value of $d_{2}$ is 3. Therefore
\[\begin{array}{r@{\hskip0pt}l}\displaystyle d_{k+1}& \displaystyle \ge d_{k}(d_{k}-1)+1\ge \big(d_{k-1}(d_{k-1}-1)+1\big)\big(d_{k-1}(d_{k-1}-1)+1\big)\\{} & \displaystyle \ge {\big(d_{k-1}(d_{k-1}-1)\big)}^{2}+1\\{} & \displaystyle \ge {\big(\big(d_{k-2}(d_{k-2}-1)+1\big)\big(d_{k-2}(d_{k-2}-1)+1-1\big)\big)}^{2}+1\\{} & \displaystyle \ge {\big(d_{k-2}(d_{k-2}-1)\big)}^{4}\ge {\big(d_{k-3}(d_{k-3}-1)\big)}^{{2}^{3}}\ge \cdots \\{} & \displaystyle \ge {\big(d_{k-(k-2)}(d_{k-(k-2)}-1)\big)}^{{2}^{k-2}}={\big(d_{2}(d_{2}-1)\big)}^{{2}^{k-2}}\ge 3\cdot {2}^{{2}^{k-2}}.\end{array}\]
So for the Sylvester series:
It is clear that
Therefore, for λ-almost all $x\in [0,1]$ their difference Sylvester expansion contain arbitrary digit i only finitely many times.
Example 5.
Consider the case where $a_{n}=d_{n}$, $b_{n}=1$. Then
\[\begin{array}{r@{\hskip0pt}l}\displaystyle d_{n+1}& \displaystyle \ge d_{n}\cdot d_{n}(d_{n}-1)+1\ge {d_{n}^{2}}\ge {\big({d_{n-1}^{2}}\big)}^{2}\\{} & \displaystyle ={d_{n-1}^{4}}\ge {d_{n-2}^{8}}\ge {d_{n-3}^{{2}^{4}}}\ge \cdots \ge {d_{n-(n-1)}^{{2}^{n}}}={d_{1}^{{2}^{n}}}\ge {2}^{{2}^{n}}.\end{array}\]
So for λ-almost all $x\in [0,1]$ the difference expansion contains arbitrary digit i only finitely many times.
3 On singularity of distribution of random variables with independent symbols of the difference restricted Oppenheim expansion
Definition 1.
A probability measure $\mu _{\xi }$ of a random variable ξ is said to be singularly continuous (with respect to the Lebesgue measure) if $\mu _{\xi }$ is a continuous probability measure and there exists a set E, such that $\lambda (E)=0$ and $\mu _{\xi }(E)=1$.
Let $x={\varDelta _{\alpha _{1}(x)\alpha _{2}(x)\dots \alpha _{n}(x)\dots }^{\overline{\mathit{ROE}}}}$ be $\overline{\mathit{ROE}}$ of real numbers, let $\xi _{1},\xi _{2},\dots ,\xi _{k},\dots $ be a sequence of independent random variables taking values $1,2,\dots ,n,\dots $ with probabilities $p_{1k},p_{2k},\dots ,p_{nk},\dots $ correspondingly, and let
be a random variables with independent $\overline{\mathit{ROE}}$-symbols.
Theorem 3.
Let assumptions of Theorem 2 hold. If there exists a digit $i_{0}$ such that ${\sum \limits_{k=1}^{\infty }}p_{i_{0}k}=+\infty $, then the probability measure $\mu _{\xi }$ is singular with respect to the Lebesgue measure.
Proof.
Consider sets
and
It is clear, that $A_{i_{0}}=\displaystyle\varlimsup_{n\to \infty }{\bar{\bigtriangleup }_{i_{0}}^{n}}$.
From the definition of ${\bar{\bigtriangleup }_{i_{0}}^{n}}$ it follows that $\mu _{\xi }({\bar{\bigtriangleup }_{i_{0}}^{n}})=p_{i_{0}n}$.
Since the random variables $\xi _{1},\xi _{2},\dots ,\xi _{n},\dots $ are independent, we conclude that
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \mu _{\xi }\big({\bar{\bigtriangleup }_{i_{0}}^{k_{1}}}\cap {\bar{\bigtriangleup }_{i_{0}}^{k_{2}}}\cap \cdots \cap {\bar{\bigtriangleup }_{i_{0}}^{k_{s}}}\big)\\{} & \displaystyle \hspace{1em}=\mu _{\xi }\big(\big\{x:\alpha _{k_{1}}(x)=i_{0},\alpha _{k_{2}}(x)=i_{0},\dots ,\alpha _{k_{s}}(x)=i_{0}\big\}\big)\\{} & \displaystyle \hspace{1em}=\mu _{\xi }\big(\big\{x:\alpha _{k_{1}}(x)=i_{0}\big\}\big)\cdot \mu _{\xi }\big(\big\{x:\alpha _{k_{2}}(x)=i_{0}\big\}\big)\cdot \dots \cdot \mu _{\xi }\big(\big\{x:\alpha _{k_{s}}(x)=i_{0}\big\}\big)\\{} & \displaystyle \hspace{1em}=p_{i_{0}k_{1}}\cdot p_{i_{0}k_{2}}\cdot \dots \cdot p_{i_{0}k_{s}}.\end{array}\]
So, events ${\bar{\bigtriangleup }_{i_{0}}^{1}},{\bar{\bigtriangleup }_{i_{0}}^{2}},\dots ,{\bar{\bigtriangleup }_{i_{0}}^{n}},\dots $ are independent with respect to measure $\mu _{\xi }$.
Since ${\sum \limits_{k=1}^{\infty }}p_{i_{0}k}=+\infty $ and $\{{\bar{\bigtriangleup }_{i_{0}}^{n}}\}$ is a sequence of independent events, from the Borel–Cantelli Lemma it follows that
Let λ be the Lebesgue measure. Events ${\bar{\bigtriangleup }_{i_{0}}^{1}},{\bar{\bigtriangleup }_{i_{0}}^{2}},\dots ,{\bar{\bigtriangleup }_{i_{0}}^{n}},\dots $, in general, are not independent w.r.t. the Lebesgue measure. We estimate the Lebesgue measure of the set ${\bar{\bigtriangleup }_{i_{0}}^{n}}$:
\[\begin{array}{r@{\hskip0pt}l}& \displaystyle \lambda \big({\bar{\bigtriangleup }_{i_{0}}^{n}}\big)=\lambda \big(\big\{x:\alpha _{n}(x)=i_{0}\big\}\big)\\{} & \displaystyle \hspace{1em}={\sum \limits_{\alpha _{1}(x)=1}^{\infty }}{\sum \limits_{\alpha _{2}(x)=1}^{\infty }}\cdots {\sum \limits_{\alpha _{n-1}=1}^{\infty }}\big|{\varDelta _{\alpha _{1}(x)\alpha _{2}(x)\dots \alpha _{n-1}(x)i_{0}}^{\overline{\mathit{ROE}}}}\big|\\{} & \displaystyle \hspace{1em}={\sum \limits_{\alpha _{1}(x)=1}^{\infty }}{\sum \limits_{\alpha _{2}(x)=1}^{\infty }}\cdots {\sum \limits_{\alpha _{n-1}=1}^{\infty }}\frac{|{\varDelta _{\alpha _{1}(x)\alpha _{2}(x)\dots \alpha _{n-1}(x)i_{0}}^{\overline{\mathit{ROE}}}}|}{|{\varDelta _{\alpha _{1}(x)\alpha _{2}(x)\dots \alpha _{n-1}(x)}^{\overline{\mathit{ROE}}}}|}\cdot \big|{\varDelta _{\alpha _{1}(x)\alpha _{2}(x)\dots \alpha _{n-1}(x)}^{\overline{\mathit{ROE}}}}\big|\\{} & \displaystyle \hspace{1em}\le l_{n}(i_{0})\cdot {\sum \limits_{\alpha _{1}(x)=1}^{\infty }}{\sum \limits_{\alpha _{2}(x)=1}^{\infty }}\cdots {\sum \limits_{\alpha _{n-1}=1}^{\infty }}\big|{\varDelta _{\alpha _{1}(x)\alpha _{2}(x)\dots \alpha _{n-1}(x)}^{\overline{\mathit{ROE}}}}\big|=l_{n}\cdot 1,\end{array}\]
where $l_{n}$ are defined in Theorem 2. Therefore,
So by the Borel–Cantelli Lemma, $\lambda (A_{i_{0}})=0$, i.e. for λ-almost all $x\in [0,1]$ their $\overline{\mathit{ROE}}$ contains arbitrary digit i only finitely many times.
Hence $\lambda (A_{i_{0}})=0$, and $\mu _{\xi }(A_{i_{0}})=1$. So, probability measure $\mu _{\xi }$ is singular with respect to the Lebesgue measure □
Theorem 4.
Let assumptions of Theorem 2 hold. If $\xi _{k}$ are independent and identically distributed random variables, then the probability measure $\mu _{\xi }$ is singular with respect to the Lebesgue measure.
Proof.
If $\xi _{1},\xi _{2},\dots ,\xi _{n},\dots $ are independent and identically distributed random variables, then $p_{ik}=p_{i}$.
Since ${\sum \limits_{i=1}^{\infty }}p_{i}=1$, it is clear that there exists a number $i_{0}$ such, that: $p_{i_{0}}>0$. Therefore
and the singularity of $\mu _{\xi }$ follows directly from Theorem 3. □
Corollary 1.
Let
be the difference version of the Sylvester expansion $(\overline{S}$-expansion) and let
be the random variable with independent symbols of $\overline{S}$-expansion.
If there exists a digit $i_{0}$ such that ${\sum \limits_{k=1}^{\infty }}p_{i_{0}k}=+\infty $, then the probability measure $\mu _{\xi }$ is singular with respect to the Lebesgue measure.
In particular, the distribution of the random variable with independent identically distributed symbols of $\overline{S}$-expansion is singular w.r.t. the Lebesgue measure.