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Topological and metric properties of sets of real numbers with conditions on their expansions in Ostrogradskii series

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Ukrainian Mathematical Journal Aims and scope

Abstract

We study topological and metric properties of the set

$$c[\bar o^1 ,\{ V_n \} ] = \left\{ {x:x = \sum\limits_n {\frac{{( - 1)^{n - 1} }}{{g_1 (g_1 + g_2 ) \ldots (g_1 + g_2 + \ldots + g_n )}},g_k \in V_k \subset \mathbb{N}} } \right\}$$

with certain conditions on the sequence of sets {V n }. In particular, we establish conditions under which the Lebesgue measure of this set is (a) zero and (b) positive. We compare the results obtained with the corresponding results for continued fractions and discuss their possible applications to probability theory.

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References

  1. F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Oxford University, New York (1995).

    MATH  Google Scholar 

  2. W. Sierpiński, “Sur quelques algorithmes pour développer les nombres réels en séries,” in: Oeuvres Choisies, Vol. 1, PWN, Warsaw (1974), pp. 236–254.

    Google Scholar 

  3. M. V. Prats’ovytyi, Fractal Approach in the Investigation of Singular Distributions [in Ukrainian], Drahomanov National Pedagogic University, Kyiv (1998).

    Google Scholar 

  4. E. Ya. Remez, “On alternating series that may be associated with two Ostrogradskii algorithms for the approximation of irrational numbers,” Usp. Mat. Nauk, 6, No. 5 (45), 33–42 (1951).

    MATH  MathSciNet  Google Scholar 

  5. M. Stern, “Über Irrationalität des Werthes gewisser Reihen,” J. Reine Angew. Math., 37, 95–96 (1848).

    MATH  Google Scholar 

  6. T. A. Pierce, “On an algorithm and its use in approximating roots of algebraic equations,” Amer. Math. Mon., 36, 523–525 (1929).

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Paradís, P. Viader, and L. Babiloni, “A mathematical excursion: from the three-door problem to a Cantor-type set,” Amer. Math. Mon., 106, 241–251 (1999).

    Article  MATH  Google Scholar 

  8. Yu. V. Mel’nichuk, “p-Adic continued fractions formed according to Euclidean and Ostrogradskii algorithms,” in: Proceedings of the Scientific Conference “Computational Mathematics in Contemporary Scientific and Technological Progress, 1974” [in Russian], Kanev (1974), pp. 259–265.

  9. Yu. V. Mel’nichuk, “On representation of real numbers by rapidly convergent series,” in: Continued Fractions and Their Applications [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1976), pp. 77–78.

    Google Scholar 

  10. Yu. V. Melnichuk, “Fast converging series representations of real numbers and their implementations in digital processing,” in: Computational Number Theory (Kossuth Lajos University, Debrecen, Hungary, September 4–9, 1989), de Gruyter, Berlin (1991), pp. 27–29.

    Google Scholar 

  11. P. I. Bondarchuk and V. Ya. Skorobohat’ko, Branching Continued Fractions and Their Applications [in Ukrainian], Naukova Dumka, Kyiv (1974).

    Google Scholar 

  12. K. G. Valeev and E. D. Zlebov, “On the metric theory of the Ostrogradskii algorithm,” Ukr. Mat. Zh., 27, No. 1, 64–69 (1975).

    Article  MathSciNet  Google Scholar 

  13. J. O. Shallit, “Some predictable Pierce expansions,” Fibonacci Quart., 22, No. 4, 332–335 (1984).

    MATH  MathSciNet  Google Scholar 

  14. P. Erdős and J. O. Shallit, “New bounds on the length of finite Pierce and Engel series,” Sémin. Théor. Nombers Bordeaux, Ser. 2, 3, No. 1, 43–53 (1991).

    Google Scholar 

  15. A. Khopfmacher and M. E. Mays, “Pierce expansions of ratios of Fibonacci and Lucas numbers and polynomials,” Fibonacci Quart., 33, No. 2, 153–163 (1995).

    MathSciNet  Google Scholar 

  16. J. Paradís, P. Viader, and L. Babiloni, “Approximation of quadratic irrationals and their Pierce expansions,” Fibonacci Quart., 36, No. 2, 146–153 (1985).

    Google Scholar 

  17. P. Viader, J. Paradís, and L. Babiloni, “Note on the Pierce expansion of a logarithm,” Fibonacci Quart., 37, No. 3, 198–202 (1999).

    MATH  MathSciNet  Google Scholar 

  18. J. O. Shallit, “Metric theory of Pierce expansions,” Fibonacci Quart., 24, No. 1, 22–40 (1986).

    MATH  MathSciNet  Google Scholar 

  19. P. Viader, L. Babiloni, and J. Paradís, “On a problem of Alfréd Rényi,” Acta Arithm., 91, No. 2, 107–115 (1999).

    MATH  Google Scholar 

  20. J. Paradís, P. Viader, and L. Babiloni, “A total order in (0juvy 1] defined through a ‘next’ operator,” Order, 16, No. 3, 207–220 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  21. A. Khopfmacher and J. Khopfmacher, “Two constructions of the real numbers via alternating series,” Int. J. Math. Math. Sci., 12, No. 3, 603–613 (1989).

    Article  Google Scholar 

  22. J. Shallit, “Pierce expansions and rules for the determination of leap years,” Fibonacci Quart., 32, No. 5, 416–423 (1994).

    MATH  MathSciNet  Google Scholar 

  23. J. Paradís, L. Babiloni, and P. Viader, “On actually computable bijections between ℕ and ℚ+,” Order, 13, No. 4, 369–377 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  24. O. M. Baranovs’kyi, “Some problems of the metric theory of numbers represented by Ostrogradskii series of the first kind,” Nauk. Zap. Drahomanov Nats. Ped. Univ., Ser. Fiz.-Mat., No. 3, 391–402 (2002).

  25. M. V. Prats’ovytyi and O. M. Baranovs’kyi, “Properties of distributions of random variables with independent differences of successive elements of Ostrogradskii series,” Teor. Imovir. Mat. Statist., No. 70, 131–144 (2004).

  26. M. V. Prats’ovytyi and O. M. Baranovs’kyi, “On the Lebesgue measure of certain sets of numbers defined by properties of their expansion in Ostrogradskii series,” Nauk. Chasop. Drahomanov Nats. Ped. Univ., Ser. 1. Fiz.-Mat., No. 5, 217–227 (2004).

  27. S. Albeverio, O. Baranovskyi, M. Pratsiovytyi, and G. Torbin, The Ostrograsky Series and Related Probability Measures, Preprint SFB-611, Bonn University, Bonn (2006); arXiv:math.PR/0605747 (2006).

    Google Scholar 

  28. A. Ya. Khinchin, Continued Fractions [in Russian], Fizmatgiz, Moscow (1961).

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kyiv

    O. M. Baranovs’kyi, M. V. Prats’ovytyi & H. M. Torbin

  2. National Pedagogic University, Kyiv

    O. M. Baranovs’kyi, M. V. Prats’ovytyi & H. M. Torbin

  3. Institut für Angewandte Mathematik, Universität Bonn, Germany

    H. M. Torbin

Authors
  1. O. M. Baranovs’kyi
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  2. M. V. Prats’ovytyi
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  3. H. M. Torbin
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__________

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 9, pp. 1155–1168, September, 2007.

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Baranovs’kyi, O.M., Prats’ovytyi, M.V. & Torbin, H.M. Topological and metric properties of sets of real numbers with conditions on their expansions in Ostrogradskii series. Ukr Math J 59, 1281–1299 (2007). https://doi.org/10.1007/s11253-007-0088-y

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  • DOI: https://doi.org/10.1007/s11253-007-0088-y

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