Weighted real Egyptian numbers (Q2422596)

The present article is devoted to certain topological aspects of representations of real numbers by finite sums of certain fractions (numerators and denominators of these fractions are positive real numbers). The author notes that results obtained in this article generalize Sierpiński's investigations from the paper [\textit{W. Sierpiński}, Mathesis 65, 16--32 (1956; Zbl 0075.03302), reproduced in Œuvres choisies. Tome I: Bibliographie, théorie des nombres et analyse mathématique. Warszawa: PWN - Editions Scientifiques de Pologne (1974; Zbl 0285.01022), p. 160--184]. In the last-mentioned paper, Sierpiński considered Egyptian numbers, i.e., rational numbers represented by finite sums of certain fractions (numerators of these fractions equal $1$). Also, in the present article, the author notes Richard K. Guy's book, that is devoted to unsolved problems in number theory, and in which were noted open problems related with Egyptian fractions. The main attention is given to some properties of the set of all weighted real Egyptian numbers (expansions of elements of this set are positive), of the set of all signed weighted Egyptian numbers (expansions of elements of this set are sign-variable), and of certain representation functions related with these sets.