Vasylenko N. The use of Fibonacci numeration system for study of fractal properties of mathematical objects with complex local structure

We introduce two expansions (representations) of real numbers in the form of subsums of series related to Fibonacci sequences: 1) terms of series are the elements of infinitesimal Fibonacci sequence; 2) terms of series are the reciprocal elements of classical Fibonacci sequence. We study the geometry of these representation systems (geometric meaning of numerals, properties of cylindrical sets, peculiarity of their intersections, basic metric relation). The problem about the number of representations of real number is solved completely. The most "economical'' representation (canonical representation) is defined. Sufficient conditions and criterions of canonicity of representation are given. Topological, metric and fractal properties of sets of numbers with conditions on digits are described.