Mykytyuk I. Non-traditional numeration systems and related fractals

In the thesis we study non-traditional numeration systems for real numbers: s-m adic system (expansions of numbers by powers of positive integer with digits), binary-cylindrical representation (in the form of subsums of given convergent series of positive terms), decimal system with binary alphabet. For each of specified numeration systems we examine the geometry of representation of numbers (properties of cylindrical sets, metric relations, geometric meaning of numerals). The question about the number of representation of real number is studied exhaustively. Some problems of metric, probabilistic and fractal theory of numbers are solved. We describe topological, metric and fractal properties of sets of numbers with conditions on digits. Properties of random variable with independent discrete symbols in some numeration system are also studied. In particular, we investigate the asymptotic behaviour of the absolute value of the characteristic function of this random variable at infinity.