The Ostrogradsky series and related Cantor-like sets

Abstract

We develop a metric and probabilistic theory for the Ostrogradsky representation of real numbers, i.e., the expansion of a real number $x$ in the following form: \begin{align*} x&= \sum_n\frac{(-1)^{n-1}}{q_1q_2... q_n}= &=\sum_n\frac{(-1)^{n-1}}{g_1(g_1+g_2)...(g_1+g_2+...+g_n)}\equiv \bO1(g_1,g_2,...,g_n,...), \end{align*} where $q_{n+1}>q_n\in\N$, $g_1=q_1$, $g_{k+1}=q_{k+1}-q_k$. We compare this representation with the corresponding one in terms of continued fractions. We establish basic metric relations (equalities and inequalities for ratios of the length of cylindrical sets). We also compute the Lebesgue measure of subsets belonging to some classes of closed nowhere dense sets defined by characteristic properties of the $\bO1$-representation. In particular, the conditions for the set $\Cset{V}$, consisting of real numbers whose $\bO1$-symbols take values from the set $V \subset N$, to be of zero resp. positive Lebesgue measure are found. For a random variable $\xi$ with independent $\bO1$-symbols $g_n(\xi)$ we prove the theorem establishing the purity of the distribution. In the case of singularity the conditions for such distributions to be of Cantor type are also found.