Geodesic
Expansions of numbers in positive L\"uroth series and their applications to metric, probabilistic and fractal theories of numbers
Algebra and discrete mathematics, Tome 14 (2012) no. 1, pp. 145-160

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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
Mots-clés : fractal, Hausdorff–Besicovitch dimension. 
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     author = {Yulia Zhykharyeva and Mykola Pratsiovytyi},
     title = {Expansions of numbers in positive {L\"uroth} series and their applications to metric, probabilistic and fractal theories of numbers},
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Yulia Zhykharyeva; Mykola Pratsiovytyi. Expansions of numbers in positive L\"uroth series and their applications to metric, probabilistic and fractal theories of numbers. Algebra and discrete mathematics, Tome 14 (2012) no. 1, pp. 145-160. http://geodesic.mathdoc.fr/item/ADM_2012_14_1_a10/