2017
Том 69
№ 9

All Issues

Topological and metric properties of sets of real numbers with conditions on their expansions in Ostrogradskii series

Baranovskyi O. M., Pratsiovytyi M. V., Torbin H. M.

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Abstract

We study topological and metric properties of the set $$C\left[\overline{O}^1, \{V_n\}\right] = \left\{x:\; x= ∑_n \frac{(−1)^{n−1}}{g_1(g_1 + g_2)…(g_1 + g_2 + … + g_n)},\quad g_k ∈ V_k ⊂ \mathbb{N}\right\}$$ with certain conditions on the sequence of sets $\{V_n\}$. In particular, we establish conditions under which the Lebesgue measure of this set is (a) zero and (b) positive. We compare the results obtained with the corresponding results for continued fractions and discuss their possible applications to probability theory.

English version (Springer): Ukrainian Mathematical Journal 59 (2007), no. 9, pp 1281-1299.

Citation Example: Baranovskyi O. M., Pratsiovytyi M. V., Torbin H. M. Topological and metric properties of sets of real numbers with conditions on their expansions in Ostrogradskii series // Ukr. Mat. Zh. - 2007. - 59, № 9. - pp. 1155–1168.

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