##
**Singularity of the second Ostrogradskii random series.**
*(Ukrainian, English)*
Zbl 1224.11070

Teor. Jmovirn. Mat. Stat. 81, 164-172 (2009); translation in Theory Probab. Math. Stat. 81, 187-195 (2010).

The authors deal with the second M. V. Ostrogradskii expansion of a positive real numbers into sign-alternating series (see E. Ya. Remez, Usp. Mat. Nauk 6, No. 5(45), 33–42 (1951; Zbl 0045.02102)):
\[
x=\sum_k\frac{(-1)^{k-1}}{q_k(x)},\quad q_k(x)\in\mathbb N,q_{k+1}(x)\geq q_k(x)(q_k(x)+1),
\]
which is called the \(O2\)-representation of \(x\) and is denoted by \(O2(q_1(x),\dots, q_n(x),\dots)\). The Ostrogradskii series converges quickly and this allows one to approximate irrational numbers effectively by partial sums of their Ostrogradskii series. Note that the partial sums are rational numbers. Let \(d_1 = q_1\) and \(d_{k+1}=q_{k+1}-q_k(q_k+1)+1\) for all \(k\in\mathbb N\). Then the latter series can be rewritten as follows:
\[
x=\sum_k\frac{(-1)^{k+1}}{q_{k-1}(x)(q_{k-1}(x)+1)-1+d_k(x)}:=\overline{O2}(d_1(x),d_2(x),\dots,d_k(x),\dots).
\]

This expression is called the \(\overline{O2}\)-expansion (the second Ostrogradskii difference expansion or the second Ostrogradskii expansion with independent increments), and the number \(d_k = d_k(x)\) is called the \(k\)-th \(\overline{O2}\)-symbol of the number \(x\). In a \(\overline{O2}\)-representation, any symbol, independently of the values of the preceding one, may attain every natural value.

The contributions of the authors represented in this paper are:

1) Developing the ergodic theory of \(\overline{O2}\)-expansions of real numbers expressed in terms of asymptotic frequencies \(\nu(x, \overline{O2})\) of \(\overline{O2}\)-symbols; in particular, finding normal properties of real numbers (properties that hold for almost all real numbers with respect to Lebesgue measure).

2) Investigating properties of the dynamical system generated by the one-sided shift transformation \(T\) in the \(\overline{O2}\)-representation.

3) Studying properties of the random variable \(\eta=\overline{O2}(\eta_1,\eta_2,\dots,\eta_k,\dots)\), where \(\eta_k\) are independent identically distributed random variables with the distribution \(P\{\eta_k=j\}=p_j\), \(j=1,2\dots\), \(p_j\geq0\), \(\sum_{j=1}^{\infty}p_j=1\). They completely describe the Lebesgue structure of this distribution. In particular, it is proved that it cannot be absolutely continuous.

The ergodic theory for the second Ostrogradskii expansion is developed. One of the results is that, for almost all (in the sense of Lebesgue measure) real numbers of the unit interval, an arbitrary symbol of an alphabet occurs finitely often in the corresponding Ostrogradskii difference expansion. It is shown that there is no probability measure that is invariant and ergodic with respect to the one-sided shift transformations \(T\) of the Ostrogradskii difference representation and absolutely continuous with respect to Lebesgue measure.

This expression is called the \(\overline{O2}\)-expansion (the second Ostrogradskii difference expansion or the second Ostrogradskii expansion with independent increments), and the number \(d_k = d_k(x)\) is called the \(k\)-th \(\overline{O2}\)-symbol of the number \(x\). In a \(\overline{O2}\)-representation, any symbol, independently of the values of the preceding one, may attain every natural value.

The contributions of the authors represented in this paper are:

1) Developing the ergodic theory of \(\overline{O2}\)-expansions of real numbers expressed in terms of asymptotic frequencies \(\nu(x, \overline{O2})\) of \(\overline{O2}\)-symbols; in particular, finding normal properties of real numbers (properties that hold for almost all real numbers with respect to Lebesgue measure).

2) Investigating properties of the dynamical system generated by the one-sided shift transformation \(T\) in the \(\overline{O2}\)-representation.

3) Studying properties of the random variable \(\eta=\overline{O2}(\eta_1,\eta_2,\dots,\eta_k,\dots)\), where \(\eta_k\) are independent identically distributed random variables with the distribution \(P\{\eta_k=j\}=p_j\), \(j=1,2\dots\), \(p_j\geq0\), \(\sum_{j=1}^{\infty}p_j=1\). They completely describe the Lebesgue structure of this distribution. In particular, it is proved that it cannot be absolutely continuous.

The ergodic theory for the second Ostrogradskii expansion is developed. One of the results is that, for almost all (in the sense of Lebesgue measure) real numbers of the unit interval, an arbitrary symbol of an alphabet occurs finitely often in the corresponding Ostrogradskii difference expansion. It is shown that there is no probability measure that is invariant and ergodic with respect to the one-sided shift transformations \(T\) of the Ostrogradskii difference representation and absolutely continuous with respect to Lebesgue measure.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

### MSC:

11K55 | Metric theory of other algorithms and expansions; measure and Hausdorff dimension |

60G30 | Continuity and singularity of induced measures |

37B10 | Symbolic dynamics |

37A50 | Dynamical systems and their relations with probability theory and stochastic processes |