This article is cited in 1 paper

RESEARCH ARTICLE

Expansions of numbers in positive Lüroth series and their applications to metric, probabilistic and fractal theories of numbers

Yulia Zhykharyeva, Mykola Pratsiovytyi

Physics and Mathematics Institute, Dragomanov National Pedagogical University, Pyrogova St. 9, 01601 Kyiv, Ukraine

Abstract: We describe the geometry of representation of numbers belonging to $(0,1]$ by the positive Lüroth series, i.e., special series whose terms are reciprocal of positive integers. We establish the geometrical meaning of digits, give properties of cylinders, semicylinders and tail sets, metric relations; prove topological, metric and fractal properties of sets of numbers with restrictions on use of “digits”; show that for determination of Hausdorff–Besicovitch dimension of Borel set it is enough to use connected unions of cylindrical sets of the same rank. Some applications of $L$-representation to probabilistic theory of numbers are also considered.

Keywords: Lüroth series, $L$-representation, cylinder, semicylinder, shift operator, random variable defined by $L$-representation, fractal, Hausdorff–Besicovitch dimension.

MSC: 11K55

Received: 02.07.2012
Accepted: 02.07.2012

Language: English

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