Maslova Y. Topologo-metric and fractal theory of the bibased G_2-representation of numbers and its application

The thesis belongs to the field of metric number theory. In the work, we develop topological, metric, probabilistic, and fractal theory of real numbers based on two-symbol system of encoding of numbers (G_2-representation). This representation is an analogue of the known two-base system (Q_2-representation). Both bases of the latter system are positive but, for the former system, one base is positive and other base is negative. For the latter system, we give new applications and, for the former system, we create a new complete theory having some essential differences. Both representations are of the same type formally, however they are not topologically equivalent. They have the same basic metric relations and similar metric component of the theory but some special functions (left and right shift operators, inversor) have essentially different properties. We prove that the left shift operator is continuous, the inversor of digits of the representation is everywhere discontinuous nowhere monotonic function and the right shift operators are linear functions with different monotonicity and the same values at point g_0. This is an essential difference between the G_2-representation and other known two-symbol representations, particularly nega-binary representation. In the thesis, we propose a generalization of Rademacher and Walsh functions based on the Q_2-representation of numbers. Their integral properties are studied. We prove that generalized Rademacher functions form an orthogonal system of functions. For every generalized Walsh function, its analytical expression is found. We give a generalization of continuous non-differentiable Bush, Wunderlich, and Tribin functions based on Q_s^*-representation of numbers of closed interval [0,1] such that properties of continuity, nowhere monotonicity, and self-similarity are preserved. Its variational properties are studied. We also describe the properties of level sets, particularly their massivity.