Geometry of numerical series: a series as a model of a real number in a new two-character number coding system

Abstract

In the paper, we consider sets of incomplete sums of subharmonic normalized series. Their topological, metric, and fractal properties are studied. We give model examples and prove theorems of existence as well as mas- siveness of classes of series. A new polybase two-symbol system of encoding of real numbers with zero redundancy is also introduced. It is topologically equivalent to representation of numbers by ????2 -continued fractions and by nega-binary representation. For this system, we describe geometry (properties of cylindrical sets) and foundations of metric theory.