Karvatsky D. Representation of real numbers by generalized Fibonacci sequences and their applications

In the thesis, we consider the properties of generalized Fibonacci sequences and their applications, use this sequences for representation of real numbers, construct a corresponding metric and probabilistic theory of numbers as well as define mathematical objects with difficult local structure (fractal sets, singular functions, distributions of random variables). We consider two analytic systems of encoding of real numbers with two-symbol and three-symbol alphabet. First system is based on representation of real number by subsum (incomplete sum) of a given series which terms are elements of infinitesimal positive generalized Fibonacci sequence. Second system of representation for real numbers based on their expansions in series such that its basic sequence is a sequence of reciprocal terms of Jacobsthal-Lucas sequence