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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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. https://geodesic-test.mathdoc.fr/item/ADM_2012_14_1_a10/

[1] S. Albeverio, O. Baranovskyi, M. Pratsiovytyi, G. Torbin, “The Ostrogradsky series and related Cantor-like sets”, Acta Arith., 130:3 (2007), 215–230 | DOI | MR

[2] Ukrainian Math. J., 59:9 (2007), 1281–1299 | DOI | MR

[3] J. Barrionuevo, R. M. Burton, K. Dajani, C. Kraaikamp, “Ergodic properties of generalized Lüroth series”, Acta Arith., 74:4 (1996), 311–327 | MR | Zbl

[4] K. Dajani, C. Kraaikamp, “On approximation by Lüroth series”, J. Théor. Nombres Bordeaux, 8:2 (1996), 331–346 | DOI | MR | Zbl

[5] C. Ganatsiou, “On some properties of the Lüroth-type alternating series representations for real numbers”, Int. J. Math. Math. Sci., 28:6 (2001), 367–373 | DOI | MR | Zbl

[6] S. Kakutani, “On equivalence of infinite product measures”, Ann. of Math., 49 (1948), 214–224 | DOI | MR | Zbl

[7] S. Kalpazidou, A. Knopfmacher, J. Knopfmacher, “Metric properties of alternating Lüroth series”, Portugal. Math., 48:3 (1991), 319–325 | MR | Zbl

[8] J. Lüroth, “Ueber eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe”, Math. Ann., 21:3 (1883), 411–423 | DOI | MR

[9] I. M. Pratsiovyta, “Second Ostrogradsky series and distributions of their random incomplete sums”, Trans. Dragomanov Nat. Pedagogical Univ. Ser. 1, Physics Mathematics, 2006, no. 7, 174–189 (in Ukrainian)

[10] I. M. Pratsiovyta, M. V. Zadniprianyi, “Expansions of numbers in Sylvester series and their applications”, Trans. Dragomanov Nat. Pedagogical Univ. Ser. 1, Physics Mathematics, 2009, no. 10, 73–87 (in Ukrainian)

[11] M. V. Pratsiovytyi, Fractal approach to investigations of singular probability distributions, Dragomanov Nat. Pedagogical Univ. Publ., Kyiv, 1998 (in Ukrainian)

[12] M. V. Pratsiovytyi, B. I. Hetman, “Engel series and their applications”, Trans. Dragomanov Nat. Pedagogical Univ. Ser. 1, Physics Mathematics, 2006, no. 7, 105–116 (in Ukrainian)

[13] M. V. Pratsiovytyi, Yu. V. Khvorostina, “Set of incomplete sums of alternating Lüroth series and probability distributions on it”, Trans. Dragomanov Nat. Pedagogical Univ. Ser. 1, Physics Mathematics, 2009, no. 10, 14–27 (in Ukrainian)

[14] Theory Probab. Math. Statist., 1998, no. 57, 143–148 | MR

[15] W. Sierpiński, “O kilku algorytmach dla rozwijania liczb rzeczywistych na szeregi”, Sprawozdania z posiedzeń Towarzystwa Naukowego Warszawskiego, Wydział III, 4 (1911), 56–77

[16] J. J. Sylvester, “On a point in the theory of vulgar fractions”, Amer. J. Math., 3:4 (1880), 332–335 ; Postscript, ibid. 388–389 | DOI | MR | MR

[17] Yu. I. Zhykharyeva, M. V. Pratsiovytyi, “Representation of numbers by positive Lüroth series: elements of metric theory”, Trans. Dragomanov Nat. Pedagogical Univ. Ser. 1, Physics Mathematics, 2008, no. 9, 200–211 (in Ukrainian)

[18] Yu. I. Zhykharyeva, M. V. Pratsiovytyi, “Properties of distribution of the random variable with independent symbols of positive Lüroth series representation”, Trans. IAMM NAS Ukraine, 23 (2011), 71–83 (in Ukrainian)