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A set of incomplete sums (subsums) of a absolutely convergent series has been featured in studies since 1914, when a pioneer work in this direction was published by Japanese mathematician Soichi Kakey [7]. Since then, active research has begun in this direction. And already final in the direction of classification of existing "topological types" of sets of incomplete sums of absolutely convergent series are the works of 1988 [4, 10], where the following fact is proved: If E is the set of subsums of a positive term convergent series ∣ Σ n=1 an then E is one of the following: (i) a finite union of closed intervals, (ii) homeomorphic to the Cantor set, (iii) homeomorphic to a certain set T called Cantorval, see (1.3). Beginning in 1941, in parallel with studies of the topological properties of the sets of incomplete sums, studies of their metric properties were carried out. Metric problems (Lebesgue measure problems) are of particular relevance when the set of incomplete sums of a series is nowhere dense or is a mixture of nowhere dense sets and a union of segments. Today, the following problems remain open in the general formulation for the set of incomplete sums of a converging positive series: 1) necessary and sufficient conditions for its nowhere density; 2) necessary and sufficient conditions for its zero-dimensionality (in the sense of Lebesgue measure); 3) its fractal properties, etc. [ABSTRACT FROM AUTHOR]