Skip to main content
Springer Nature Link
Log in

The Mahler problem with nonmonotone right-hand side in the field of complex numbers

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

In the last twenty years, the exact order of approximation by zeros of the moduli of the values of the integer polynomials in a real and a complex variable was established. However, in the case of convergence of the series consisting of the right-hand sides of inequalities, the monotonicity condition for the right-hand sides in the classical Khintchine theorem can be dropped. It is shown in the present paper that, in the complex case, the monotonicity condition is also insignificant for polynomials of arbitrary degree.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. K. Mahler, “Über das Maß der Menge aller S-Zahlen,” Math. Ann. 106(1), 131–139 (1932).

    Article  MathSciNet  Google Scholar 

  2. V. G. Sprindzhuk, “On a conjecture of Mahler,” Dokl. Akad. Nauk SSSR 154(4), 783–786 (1964) [Soviet Math. Dokl. 5, 183–187 (1964)].

    MathSciNet  Google Scholar 

  3. V. G. Sprindzhuk, The Mahler Problem in the Metric Theory of Numbers (Nauka i Tekhnika, Minsk, 1967) [in Russian].

    Google Scholar 

  4. V. I. Bernik and D. V. Vasil’ev, “Khintchine-type theorem for integral polynomials of a complex variable,” Tr. Inst. Mat. NAN Belarus 3, 10–20 (1999). [in Russian].

    MathSciNet  MATH  Google Scholar 

  5. A. I. Khintchine, “Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen,” Math. Ann. 92(1–2), 115–125 (1924).

    Article  MathSciNet  MATH  Google Scholar 

  6. V. V. Beresnevich, “On a theorem of V. Bernik in the metric theory of Diophantine approximation,” Acta Arith. 117(1), 71–80 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  7. V. I. Bernik, D. Kleinbock, and G. A. Margulis, “Khintchine-type theorems on manifolds: the convergence case for standart and multiplicative versions,” Int. Math. Res. Notices 9, 453–486 (2001).

    Article  MathSciNet  Google Scholar 

  8. V. I. Bernik, “The exact order of approximating zero by values of integral polynomials,” Acta Arith. 53(1), 17–28 (1989).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Khabarovsk Branch, Institute of Applied Mathematics, Far-Eastern Branch, Russian Academy of Sciences Dublin Institute of Technology, Dublin, Ireland

    N. V. Budarina

Authors
  1. N. V. Budarina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. V. Budarina.

Additional information

Original Russian Text © N. V. Budarina, 2013, published in Matematicheskie Zametki, 2013, Vol. 93, No. 6, pp. 812–820.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Budarina, N.V. The Mahler problem with nonmonotone right-hand side in the field of complex numbers. Math Notes 93, 802–809 (2013). https://doi.org/10.1134/S0001434613050192

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434613050192

Keywords

Search

Navigation