Generalizations of certain representations of real numbers

Abstract

In the present paper,  real number representations that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are  proven. The  theorem on the representation of real numbers from a certain interval is formulated.

One of  the peculiarities of  the  research presented in this paper, is introducing numeral systems with mixed bases
(i.e., with bases containing positive and negative numbers).
In 2016, an idea of a corresponding  analytic representation of numbers
was  presented in
\cite[Serbenyuk, S.: \textit{On~some generalizations of~real numbers representations}, arXiv:1602.07929v1]{S.Serbenyuk}.
These investigations were presented in
\cite[Serbenyuk, S.: \textit{Generalizations of~certain representations of real numbers},
 arXiv:1801.10540]{S.Serbenyuk2018} in January 2018.

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Also, an idea of such investigations was presented by the author of this paper at the conference  in 2015
(see \cite[Serbenyuk, S.: {\textit{Quasi-nega-$\widetilde Q$-representation as a generalization of a representation of real numbers by certain sign-variable series},
\url{https://www.researchgate.net/publication/303255656}}]{Serbenyukabstract2015}).