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Representation of real numbers by the alternating Cantor series. (English) Zbl 1412.11014
Summary: The article is devoted to alternating Cantor series. It is proved that any real number belonging to $$[a_0-1, a_0]$$, where $$a_0=\sum^{\infty}_{k=1}\frac{d_{2k}-1}{d_1d_2\cdots d_{2k}}$$, has not more than two representations by such series, and only the numbers from a certain countable subset of real numbers have two representations. The geometry of these representations, properties of cylinder and semicylinder sets, and the simplest metric problems are investigated. Some applications of such series to fractal theory and the relation between positive and alternating Cantor series are described. The shift operator with some its applications, as well as the set of incomplete sums are studied. Necessary and sufficient conditions for a rational number to be representable by an alternating Cantor series are formulated.

##### MSC:
 11A67 Other number representations 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension 28A80 Fractals
##### Keywords:
alternating Cantor series
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##### References:
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