The thesis belongs to the field of constructive theory of real functions
with locally complicated structure and fractal properties, theory of encoding
of real numbers using an infinite alphabet, metric and probability theory of
numbers, and partly appertains to the theory of dynamical systems. It is
devoted to special functions defined in terms of representation of real numbers by Engel and Ostrogradsky–Sierpi´nski–Pierce series, as well as Perron
series, which is a generalization of Engel, L¨uroth, and Sylvester series. An
important component of the thesis is a comprehensive theory of encoding of
real numbers using an infinite alphabet, which is based on the representation
of numbers by Perron series. The metric theory of dynamical systems in the
thesis is represented by the solution of an analogue of the Gauss–Kuzmin
problem (which concerned elementary continued fractions) for difference representations of numbers by Engel and Perron series.
This work is a continuation of research into various coding systems of real
numbers using an infinite alphabet and objects related to them, which were
carried out by
1) Gauss C.F., Kuzmin R.O., Khinchin A.Ya. (elementary continued
fraction);
2) Pratsiovytyi M.V., Leshchynskyi O.L., Torbin G.M., Nikiforov R.O.
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(?∞-representation);
3) Erd˝os P., R´enyi A., Galambos J., Knopfmacher A., Knopfmacher J.,
Pratsiovytyi M.V., Hetman B.I., Baranovskyi O.M. (Engel series);
4) Ostrogradsky M.V., Sierpi´nski W., Pierce T.A., Shallit J.O., Pratsiovytyi M.V., Baranovskyi O.M., Torbin G.M. (Ostrogradsky–Sierpinski–Pierce series);
5) L¨uroth J., Zhykharyeva Yu.I., Khvorostina Yu.V. (positive and alternating L¨uroth series);
6) Sylvester J.J., Erd˝os P., R´enyi A. (Sylvester series);
7) Perron O. (Perron series).
Traditional for systems of coding (representation) of numbers are questions about
1) the existence and unity of the representation;
2) the topological and metric properties of cylindrical set;
3) the asymptotic frequencies of digits;
4) the normal properties of representation;
5) the topological and metric structure of set with conditions for the use
of numbers in representation;
6) the properties of the dynamic system generated by the left-shift operator of digits.
The main objects of the research are projectors of one representation
into another representation, random variables associated with Ostrogradsky–
Sierpi´nski–Pierce and Engel series, transformation of space of representations,
dynamical systems generated by left-shift operators of digits.
