Representation of real numbers by the alternating Cantor series

Abstract

The article is devoted to the alternating Cantor series. It is proved that any real number belonging to $[a_0-1;a_0]$, where $a_0=\sum^{\infty} _{k=1} {\frac{d_{2k}-1}{d_1d_2...d_{2k}}} $, has no more than two representations by the series and only numbers from countable subset of real numbers have two representations. Geometry of the representation, properties of cylinders sets and semicylinders, simplest metric problems are investigated. Some applications of the series in fractal theory and relation between positive and alternating Cantor series are described. A shift operator and its some applications, set of incomplete sums are studied. Necessary and sufficient conditions for representations of rational numbers by the alternating Cantor series are formulated.