Fundamentals of metric theory of real numbers in their $\overline{Q_3}$-representation

I.V. Zamrii; V.V. Shkapa; H.M.  Vlasyk

doi:10.30970/ms.56.1.3-19

I.V. Zamrii State University of Telecommunication
V.V. Shkapa State University of Telecommunications, Kyiv, Ukraine
H.M. Vlasyk State University of Telecommunications, Kyiv, Ukraine

DOI: https://doi.org/10.30970/ms.56.1.3-19

Keywords: $Q_3$-representation of real numbers;, $\overline{Q_3}$-representation of real numbers;, metric properties;, cylinders;, geometry of representation;, Lebesgue measure;, Hausdorff-Besicovitch dimension;, tail set

Abstract

In the paper we were studied encoding of fractional part of a real number with an infinite alphabet (set of digits) coinciding with the set of non-negative integers. The geometry of this encoding is generated by $Q_3$-representation of real numbers, which is a generalization of the classical ternary representation. The new representation has infinite alphabet, zero surfeit and can be efficiently used for specifying mathematical objects with fractal properties.

We have been studied the functions that store the "tails" of $\overline{Q_3}$-representation of numbers and the set of such functions,
some metric problems and some problems of probability theory are connected with $\overline{Q_3}$-representation.

Author Biographies

I.V. Zamrii, State University of Telecommunication

State University of Telecommunications, Kyiv, Ukraine

V.V. Shkapa, State University of Telecommunications, Kyiv, Ukraine

State University of Telecommunications, Kyiv, Ukraine

H.M. Vlasyk, State University of Telecommunications, Kyiv, Ukraine

State University of Telecommunications, Kyiv, Ukraine

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