Dmytrenko S. Two-symbol systems of encoding of numbers related to continued fractions and their applications

The thesis is devoted to the development of topological, metric and fractal theories of real numbers based on two-symbol systems of encoding of numbers related to continued fractions and their application in metric and probability theory, fractal analysis and function theory. Two two-symbol systems of encoding of numbers with zero redundancy, related to elementary and non-elementary continued fractions (mediant representation, continued A2-representation) are substantiated. An integrated topological-metric theory of these representations has been created, the basic metric relations have been found, and some metric problems have been solved. It is proved that the mediant representation of numbers is a two-symbol recoding of the representation of numbers by elementary continued fractions. In terms of this representation a new expression of the classical singular strictly increasing Minkowski function was found. Effective applications in the theory of fractals and the theory of locally complex functions have been found for the representation of numbers by continued A2-fractions. The equivalent definition of the Hausdorff–Besicovitch fractal dimension in terms of A2-representation is found, the properties of two classes of functions are described. Functions of the first class are defined by A2-representation and infinite absolutely convergent products of numbers, and functions of the second class are set by converters of pairs of A2-digits of argument in A2-digits of value of function with their chain bond.