On the uniform distribution of remainders in the expression of real numbers over a multiplicative system numbers
https://doi.org/10.22405/2226-8383-2022-23-5-38-44
Abstract
In this paper theorems on the expression of real numbers on multiplicative number system.
It pay a special attention to “explicit formulas” and conditions of the uniqueness of such representations. Here is found that the sequence remainders in this expansion has the uniform distribution. The given statement generalises the known result of Hardy–Littlewood for a
positional system of calculus. On the base of proof lie two statement: the Weyl’s criteria of the uniform distribution of a sequence modulo unit and the theoretic-probability lemma of Borel–Cantelli.
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
3. Circolo math. Palermo, 27.
4. Gel’fond A. O. (1959). On one general property of numerical system// Izv. AN SSSR, Ser.
5. math. (in Russian). 23 (Selected works. p.366-371).
6. Zeckendorf E. (1972). Repr´sentation des nombres naturels par une somme de nombres de
7. Fibonacci ou de nombres de Lucas// Bull. Soc. R. Sci. Li`ege (in French). 41, p. 179-182.
8. Dickson L. E. (1919). History of the theory of numbers. — Carnegie Inst. of Washigton. Ch.17.
9. Arkhipov G. I., Sadovnichii V. A., Chubarikov V. N. (2006).Lectures on mathematical analysis.
10. — M.: Drofa. Pp. 640.
11. Cassels J. W. S. (1961). An introduction to Diophantine approximation. — Cambridge University
12. Press. Pp.212.
13. Hall M.,Jr. (1970). Combinatorial theory. — Waltham (Massachusetts)-Toronto-London: Blaisdell
14. Publ. Comp. Pp. 424.
15. Bernoulli D. (1728). Combinatorial theory.// Comment. Acad.Sci. Petrop., 3, p. 85–100.
16. Knuth D. E. (1998). The art computer programming. Fundamental algorithms. Third Ed. —
17. Reading, Massachusetts-Harlow, England-Menlo Park, California-Berkley, california-Lon Mills,
18. Ontario-Sidney-Bonn-Amsterdam-Tokyo-Mexico City: AddisonWesley Longman, Inc.. Pp. 720.
19. de Moivre A. (1922). // Philos. Trans., 32, p. 162–178.
20. ChebyshevP. L. (1936). The theory of probabilities. — AN SSSR. S23. 143–147. (in Russian).
21. Landau E. (1947). Fundamentals of analysis. — M.: Inostr.literature.(in Russian).
22. Golubov B. I., Efimov A. V., Skvortsov V. A. (1987). Series and the Uolsh’s transformations: the
23. theory and applications. — M.: Nauka, pp. 344.(in Russian).
24. Mineev M.P., Chubarikov V. N. (2014). Lectures on arithmetical questions of cryptography. —
25. M.: OOO“Luch”, pp. 224. (in Russian).
26. Ghyasi А. H. (2007). A generalization of the Gel’fond theorem concerning number systems//
27. Russian Journal of Mathematical Physics. 14, No.3, p.370.
Review
For citations:
Giyasi A., Mikhailov I.P., Chubarikov V.N. On the uniform distribution of remainders in the expression of real numbers over a multiplicative system numbers. Chebyshevskii Sbornik. 2022;23(5):38-44. (In Russ.) https://doi.org/10.22405/2226-8383-2022-23-5-38-44
Views:
354
Email this article 