On the sequence of fractional parts of the ratio of Fibonacci numbers 𝑥𝑛+1 = {︁((𝐹_(𝑛+1))/𝐹_𝑛)𝑥𝑛}︁
https://doi.org/10.22405/2226-8383-2023-24-3-242-250
Abstract
In this paper for the expension of real numbers on Fibonacci sequence theorems on the uniform distribution of remainders for almost of all real numbers in the sense of Lebesgue’s measure. the conclusion of this theorem is based on theWeyl’s criteria of the uniform distribution
of a sequence modulo unit and on the lemma.
About the Authors
Azar Giyasi
Allameh Tabataba’i University
Islamic Republic of Iran
Islamic Republic of Iran
candidate of physical and mathematical sciences
Ilya Petrovich Mikhailov
Kazan Aviation Institute
Russian Federation
Russian Federation
Vladimir Nikolaevich Chubarikov
Lomonosov Moscow State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
References
1. Hardy G. H., Littlewood J. E., 1914, “The fractional part of 𝑛𝑘𝜃” // Acta math., 37.
2. Borel E, 1909, “Les probabilit´es d´enombarables et leurs applications arithm´etiques” // Rend Circolo math. Palermo, 27.
3. Gel’fond A. O, 1959, “On one general property of numerical system ”// Izv. AN SSSR, Ser. math. (in Russian). 23 (Selected works) pp. 366–371.
4. Zeckendorf E, 1972, “Repr´sentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas” // Bull. Soc. R. Sci. Li`ege (in French). 41, pp. 179–182.
5. Dickson L. E, 1919, “History of the theory of numbers” — Carnegie Inst. of Washigton. Ch.17.
6. Arkhipov G. I., Sadovnichii V. A., Chubarikov V. N, 2006, “Lectures on mathematical analysis” — M.: Drofa. Pp. 640.
7. Cassels J. W. S, 1961, “An introduction to Diophantine approximation” — Cambridge University Press, pp. 212.
8. Hall M.,Jr, 1970, “Combinatorial theory” — Waltham (Massachusetts)-Toronto-London: Blaisdell Publ. Comp., pp. 424.
9. Bernoulli D, 1728, “Combinatorial theory” // Comment. Acad.Sci. Petrop., 3, pp. 85–100.
10. Knuth D. E, 1998, “The art computer programming. Fundamental algorithms. Third Ed” — Reading, Massachusetts-Harlow, England-Menlo Park, California-Berkley, california-Lon Mills, Ontario-Sidney-Bonn-Amsterdam-Tokyo-Mexico City: Addison Wesley Longman, Inc. pp. 720.
11. de Moivre A, 1922, “Philos. Trans”, 32, p. 162–178.
12. ChebyshevP. L, 1936, “The theory of probabilities” — AN SSSR. S23. pp. 143–147. (in Russian).
13. Landau E, 1947, “Fundamentals of analysis” — M.: Inostr.literature.(in Russian).
14. Golubov B. I., Efimov A. V., Skvortsov V. A, 1987, “Series and the Uolsh’s transformations: the theory and applications” — M.: Nauka, pp. 344.(in Russian).
15. Mineev M.P., Chubarikov V. N, 2014, “Lectures on arithmetical questions of cryptography” — M.: LLC “Luch”, pp. 224. (in Russian).
16. Ghyasi А. H, 2007, “A generalization of the Gel’fond theorem concerning number systems”// Russian Journal of Mathematical Physics. 14, no.3, p.370.
17. Ghyasi A. K., Mihaylov I.P., Chubarikov V. N, 2022, “On the expansion of real numbers over some sequences” // Chebyshevskii Sbornik. 23, no.3, pp.50–60.
18. Ghyasi A. K., Mihaylov I.P., ChubarikovV. N, 2022, “On the uniform distribution of remainders in the expansion of real numbers over the multiplicative system of numbers” // Chebyshevskii Sbornik. 23, No.5, с.38-44.
19. Ghyasi A. K., Mihaylov I.P., Chubarikov V. N, 2023, “On the expansion of real numbers over the Fibonacci sequence” // Chebyshevskii Sbornik. 24, no.2, pp. 247–253.
Review
For citations:
Giyasi A., Mikhailov I.P., Chubarikov V.N. On the sequence of fractional parts of the ratio of Fibonacci numbers 𝑥𝑛+1 = {︁((𝐹_(𝑛+1))/𝐹_𝑛)𝑥𝑛}︁. Chebyshevskii Sbornik. 2023;24(3):242-250. (In Russ.) https://doi.org/10.22405/2226-8383-2023-24-3-242-250
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