We present expansions of real numbers in alternating s-adic series (1 < s ∈ N), in particular, s-adic Ostrogradskii series of the first and second kind. We study the “geometry” of this representation of numbers and solve metric and probability problems, including the problem of structure and metric-topological and fractal properties of the distribution of the random variable
where τ k are independent random variables that take natural values.
Similar content being viewed by others
References
M. V. Prats’ovytyi, Fractal Approach to the Investigation of Singular Distributions [in Ukrainian], Drahomanov National Pedagogic University, Kyiv (1998).
F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Clarendon, Oxford (1995).
E. A. Remez, “On alternating series that can be associated with Ostrogradskii algorithms for the approximation of irrational numbers,” Usp. Mat. Nauk, 6, No. 5(45), 33–44 (1951).
A. Ya. Khinchin, Continued Fractions [in Russian], Fizmatgiz, Moscow (1978).
W. Sierpinski, O Kilku Algorithmach dla Rozwijania Liczb Rzeczywistych na Szeregi, Vol. 3, STNW, Warsaw (1911).
T. A. Pierce, “On an algorithm and its use in approximating roots of an algebraic equation,” Amer. Math. Monthly, 36, 523–525 (1929).
J. O. Shallit, “Pierce expansions and rules for the determination of leap years,” Fibonacci Quart., 32, No. 5, 416–423 (1994).
O. M. Baranovs’kyi, “Representation of nowhere differentiable functions using representation of numbers by Ostrogradskii series,” Fraktal. Anal. Sumizh. Pyt., No. 2, 215–221 (1998).
M. V. Prats’ovytyi and O. M. Baranovs’kyi, “Properties of distributions of random variables with independent differences of successive elements of the Ostrogradskii series,” Teor. Imovir. Mat. Statyst., 70, 131–144 (2004).
M. V. Prats’ovytyi and O. M. Baranovs’kyi, “On the Lebesgue measure of certain sets of numbers defined by properties of their expansion into Ostrogradskii series,” Nauk. Chasop. Nats. Ped. Univ. Im. Drahomanova, Ser. 1, Fiz.-Mat. Nauky, No. 5, 217–227 (2004).
I. M. Prats’ovyta, “Ostrogradskii series of the second kind and distributions of their random incomplete sums,” Nauk. Chasop. Nats. Ped. Univ. Im. Drahomanova, Ser. 1, Fiz.-Mat. Nauky, No. 7, 141–156 (2006).
S. Kakutani, “Equivalence of infinite product measures,” Ann. Math., 49, 214–224 (1948).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 7, pp. 958–968, July, 2009.
Rights and permissions
About this article
Cite this article
Prats’ovyta, I.M. On expansions of numbers in alternating s-adic series and Ostrogradskii series of the first and second kind. Ukr Math J 61, 1137–1150 (2009). https://doi.org/10.1007/s11253-009-0266-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-009-0266-1