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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
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References:
[1] G. Cantor, Ueber die einfachen Zahlensysteme, Z. Math. Phys. 14 (1869), 121-128. · JFM 02.0085.01
[2] K. Falconer, Techniques in Fractal Geometry, John Wiley & Sons, Ltd., Chichester, 1997. · Zbl 0869.28003
[3] K. Falconer, Fractal Geometry. Mathematical Foundations and Applications. Second edition, John Wiley & Sons, Inc., Hoboken, NJ, 2003.
[4] J. Galambos, Representations of Real Numbers by Infinite Series, Lecture Notes in Mathematics 502, Springer, 1976. · Zbl 0322.10002
[5] S. Ito and T. Sadahiro, Beta-expansions with negative bases, Integers 9 (2009), 239-259. · Zbl 1191.11005
[6] S. Kalpazidou, A. Knopfmacher and J. Knopfmacher, Metric properties of alternating L¨uroth series, Port. Math. 48 (1991), no. 3, 319-325. · Zbl 0735.11035
[7] B. Mance, Normal Numbers with Respect to the Cantor Series Expansion, Dissertation, The Ohio State University, 2010.
[8] S.Serbenyuk,Cantorseriesandrationalnumbers,availableat https://arxiv.org/abs/1702.00471
[9] S. O. Serbenyuk, On some sets of real numbers such that defined by nega-s-adic and Cantor nega-s-adic representations, Trans. Natl. Pedagog. Mykhailo Dragomanov Univ. Ser. 1. Phys. Math. 15 (2013), 168-187, available at https://www.researchgate.net/publication/292970280 (in Ukrainian)
[10] S. O. Serbenyuk, Real numbers representation by the Cantor series, International Conference on Algebra dedicated to 100th anniversary of S.M. Chernikov:Abstracts, Dragomanov National Pedagogical University,Kyiv,2012. - P. 136,available at https://www.researchgate.net/publication/301849984
[11] S. Serbenyuk, Nega- ˜Q-representation as
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