Cantor series expansions of rational numbers

Abstract

This article is devoted to necessary and sufficient conditions for a rational number to be representable by a Cantor series. Necessary and sufficient conditions are formulated for the case of an arbitrary sequence (q k ).

Rational numbers represented by sign-variable Cantor series

“…Cantor series expansions have been intensively studied from different points of view during the last century (for example, see a brief description in [13]). For example, the following researchers investigated problems related with Cantor series: P. Erdös, J. Galambos, G. Iommi, P. Kirschenhofer, T. Komatsu, V. Laohakosol, B. Li, M. Paštéka, S. Prugsapitak, J. Rattanamoong, A. Rényi, B. Skorulski, R. F. Tichy, P. Turán, Yi Wang, M. S. Waterman, H. Wegmann, Liu Wen, Zhixiong Wen, Lifeng Xi, and other mathematicians.…”

“…For example, G. Cantor, P. A. Diananda, A. Oppenheim, P. Erdös, J. Hančl, E.G. Straus, P. Rucki, P. Kuhapatanakul, V. Laohakosol, and other scientists studied this problem (see references in [13,15]). However, known results include different conditions satisfied by (ε k ) and/or (q k ).…”

A Note on Expansions of Rational Numbers by Certain Series

“…A number of researches are devoted to investigations of Cantor expansions from different points of view (a brief description is given in [12]) including studying the problem on Cantor series expansions of rational numbers (for example, see [1,2,7,11,12,13,14,15,17]). In [4], Prof. János Galambos calls the problem on representations of rational numbers by Cantor series (1) as the fourth open problem.…”

“…One can note that the notion of the shift operator is applicable to this problem (for example, some descritions are given in [12,15]). This paper is devoted to applications of the notion of the generalized shift operator to solving the problem on representations of rational numbers by positive Cantor series.…”

Modeling rational numbers by Cantor series

“…In 1864, G. Cantor introduced the problem on representaions of rational numbers by series (1) (see [1]). More information about this problem were described in [4].…”