Skip to main content
SpringerLink
Log in

Additive Problem with k Numbers of a Special Form

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

In this paper, we consider an additive problem of the form n1 + n2 + . . . + nk = N with at least two summands, where the summands satisfy the condition ni ∈ ℕ(αi, Ii) for 1 ≤ i ≤ k and ℕ(αI) = {n ∈ ℕ : {nα} ∈ I}.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Akiyama, “Self affine tiling and Pisot numeration system,” in: Number Theory and Its Applications (K. Gyory and S. Kanemitsu, eds.), Kluwer (1999), pp. 7–17.

  2. S. A. Gritsenko and N. N. Mot’kina, “On a version of the ternary Goldbach’s problem,” Dokl. Akad. Nauk. Tajikistan, 52, No. 6, 413–417 (2009).

    Google Scholar 

  3. S. A. Gritsenko and N. N. Mot’kina, “Hua Loo Keng’s problem involving primes of a special type,” Dokl. Akad. Nauk. Tajikistan, 52, No. 7, 497–500 (2009).

    Google Scholar 

  4. S. A. Gritsenko and N. N. Mot’kina, “On some additive problems of number theory,” Nauch. Ved. Belgorod Univ. Ser. Mat. Fiz., 5 (76), No. 18, 83–87 (2010).

  5. S. A. Gritsenko and N. N. Mot’kina, “On the computation of some singular series,” Chebyshev. Sb., 12, No. 4, 85–92 (2011).

    MathSciNet  MATH  Google Scholar 

  6. S. A. Gritsenko and N. N. Mot’kina, “On Chudakov’s theorem for prime numbers of a special form,” Chebyshev. Sb., 12, No. 4, 75–84 (2011).

    MATH  Google Scholar 

  7. S. A. Gritsenko and N. N. Mot’kina, “Warings problem involving natural numbers of a special type,” Chebyshev. Sb., 15, No. 3, 31–47 (2014).

    Google Scholar 

  8. E. P. Davletyarova, A. A. Zhukova, and A. V. Shutov, “Geometrization of Fibonacci numeration system and its applications to number theory,” Algebra Anal., 25, No. 6, 1–23 (2013).

    Google Scholar 

  9. E. P. Davletyarova, A. A. Zhukova, and A. V. Shutov, “Geometrization of the generalized Fibonacci numeration system with applications to number theory,” Chebyshev. Sb., 17, No. 2, 88–112 (2016).

    Article  MathSciNet  Google Scholar 

  10. V. V. Krasil’shchikov and A. V. Shutov, “Distribution of points of one-dimensional quasilattices with respect to a variable module,” Izv. Vyssh. Uchebn. Zaved. Mat., 3, 17–23 (2012).

    MathSciNet  MATH  Google Scholar 

  11. V. V. Krasil’shchikov, A. V. Shutov, and V. G. Zhuravlev, “One-dimensional quasiperiodic tilings admitting progressions enclosure,” Izv. Vyssh. Uchebn. Zaved. Mat., 7, 3–9 (2009).

    MATH  Google Scholar 

  12. G. Rauzy, “Nombres algbriques et substitutions,” Bull. Soc. Math. Fr., 110, 147–178 (1982).

    Article  Google Scholar 

  13. A. V. Shutov, “Numeration systems and bounded remainder sets,” Chebyshev. Sb., 7, No. 3, 110–128 (2006).

    MathSciNet  MATH  Google Scholar 

  14. A. V. Shutov, “Arithmetics and geometry of one-dimensional quasilattices,” Chebyshev. Sb., 11, No. 1, 255–262 (2010).

    MathSciNet  MATH  Google Scholar 

  15. A. V. Shutov, “Trigonometric sums over one-dimensional quasilattices,” Chebyshev. Sb., 13, No. 2, 136–148 (2012).

    MathSciNet  MATH  Google Scholar 

  16. A. V. Shutov, “On one additive problem with fractional part function,” Nauch. Ved. Belgorod Univ. Ser. Mat. Fiz., 5 (148), No. 30, 111–120 (2013).

  17. A. V. Shutov, A. V. Maleev, and V. G. Zhuravlev, “Complex quasiperiodic self-similar tilings: their parameterization, boundaries, complexity, growth and similarities,” Acta Crystal., A66, 427–437 (2010).

    Article  Google Scholar 

  18. H. Weyl, “Über die Gleichverteilung von Zahlen mod. Eins,” Math. Ann., 77 (3), 313–352 (1916).

    Article  MathSciNet  Google Scholar 

  19. A. A. Zhukova and A. V. Shutov, “Binary additive problem with numbers of special type,” Chebyshev. Sb., 16, No. 3, 247–275 (2015).

    MATH  Google Scholar 

  20. V. G. Zhuravlev, “Rauzy tilings and bounded remainder sets on the torus,” Zap. Nauchn. Sem. POMI, 322, 83–106 (2005).

    MATH  Google Scholar 

  21. V. G. Zhuravlev, “Sums of squares over the Fibonacci ○-ring,” Zap. Nauchn. Sem. POMI, 337, 165–190 (2006).

    MATH  Google Scholar 

  22. V. G. Zhuravlev, “One-dimensional Fibonacci quasilattices and their application to the Euclidean algorithm and Diophantine equations,” Algebra Anal., 19, No. 3, 151–182 (2007).

    Google Scholar 

  23. V. G. Zhuravlev, “The Pell equation over the Fibonacci ○-ring,” Zap. Nauchn. Sem. POMI, 350, 139–159 (2007).

    Google Scholar 

  24. V. G. Zhuravlev, “Even Fibonacci numbers: binary additive problem, distribution over progressions, and spectrum,” Algebra Anal., 20, No. 3, 18–46 (2008).

    MathSciNet  Google Scholar 

  25. V. G. Zhuravlev, “Hyperbolas over two-dimensional Fibonacci quasilattices,” Fundam. Prikl. Mat., 16, No. 6, 45–62 (2010).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Vladimir Branch of the Russian Presidential Academy of National Economy and Public Administration, Vladimir, Russia

    A. A. Zhukova & A. V. Shutov

  2. Vladimir State University named after Alexander and Nikolay Stoletovs, Vladimir, Russia

    A. A. Zhukova & A. V. Shutov

Authors
  1. A. A. Zhukova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. V. Shutov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. A. Zhukova or A. V. Shutov.

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 166, Proceedings of the IV International Scientific Conference “Actual Problems of Applied Mathematics,” Kabardino-Balkar Republic, Nalchik, Elbrus Region, May 22–26, 2018. Part II, 2019.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhukova, A.A., Shutov, A.V. Additive Problem with k Numbers of a Special Form. J Math Sci 260, 163–174 (2022). https://doi.org/10.1007/s10958-022-05681-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-022-05681-7

Keywords and phrases

AMS Subject Classification

Search

Navigation