Pratsiovyta I. Expansions of real numbers via the second Ostrogradsky series and their applications

The thesis is devoted to the development of metric and probabilistic theories of real numbers, represented via the second Ostrogradsky series. After fundamental study of main properties of the second Ostrogradsky series and the geometry of the corresponding expansion (geometrical meaning of "digits", properties of cylindrical sets, metric relations) a series of problems on metric, topological and fractal properties of subsets of real numbers defined via special restrictions on digits of the expansion are solved. We also compare metric and dimensional theories generated by the second Ostrogradsky expansion, by the first Ostrogradsky expansion and by the continued fractions expansion. We choose a relatively simple subclass of the second Ostrogradsky series (alternating s-adic series) and study properties of reals which can be expanded in such series. Applications of the developed metric theory in the probability theory are also considered.