Skip to main content

Advertisement

SpringerLink
Log in

Simultaneous diophantine approximation in the real and p-adic fields with nonmonotonic error function*

  • Published:
Lithuanian Mathematical Journal Aims and scope Submit manuscript

Abstract

In this paper, we show that if the volume sum \( \sum\nolimits_{h = 1}^\infty {{h^{n - 1}}{\Psi^t}(h)} \) converges for a function ψ (not necessarily monotonic), then the set of points \( \left( {x,{w_1}, \ldots, {w_{t - 1}}} \right) \in {\mathbb R} \times {{\mathbb Q}_{{p_1}}} \times \ldots \times {{\mathbb Q}_{{p_{t - 1}}}} \) that simultaneously satisfy the inequalities \( \left| {P(x)} \right| < \Psi (H) {\text{and}} {\left| {P\left( {{w_i}} \right)} \right|_{{p_i}}} < \Phi (H), 1 \leqslant i \leqslant t - 1 \), for infinitely many integer polynomials P has measure zero.

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. A. Baker, On a theorem of Sprindžuk, Proc. R. Soc. London, Ser. A, 292:92–104, 1966.

    Article  MATH  Google Scholar 

  2. V.V. Beresnevich, On approximation of real numbers by real algebraic numbers, Acta Arith., 90:97–112, 1999.

    MathSciNet  MATH  Google Scholar 

  3. V.V. Beresnevich, A Groshev-type theorem for convergence on manifolds, Acta Math. Hung., 94:99–130, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  4. 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 

  5. V.V. Beresnevich, V.I. Bernik, D.Y. Kleinbock, and G.A. Margulis, Metric Diophantine approximation: The Khintchine–Groshev theorem for nondegenerate manifolds, Mosc. Math. J., 2:203–225, 2002.

    MathSciNet  MATH  Google Scholar 

  6. V.V. Beresnevich, V.I. Bernik, and E.I. Kovalevskaya, On approximation of p-adic numbers by p-adic algebraic numbers, J. Number Theory, 111:33–56, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  7. V.I. Bernik, A metric theorem on the simultaneous approximation of zero by the values of integral polynomials, Izv. Akad. Nauk SSSR, Ser. Mat., 44:24–45, 1980. English transl.: Math. USSR, Izv., 16:21–40, 1983.

    Article  Google Scholar 

  8. V.I. Bernik, On the exact order of approximation of zero by values of integral polynomials, Acta Arith., 53:17–28, 1989.

    MathSciNet  MATH  Google Scholar 

  9. V.I. Bernik, N. Budarina, and D. Dickinson, Khintchine’s theorem and approximation of zero by the values of integer polynomials in different metrics, Dokl. Akad. Nauk, 413(2):151–153, 2007.

    MathSciNet  Google Scholar 

  10. V.I. Bernik and M.M. Dodson, Metric Diophantine Approximation on Manifolds, Cambridge Tracts Math., Vol. 137, Cambridge Univ. Press, Cambridge, 1999.

    Book  MATH  Google Scholar 

  11. V.I. Bernik, D.Y. Kleinbock, and G.A. Margulis, Khintchine-type theorems on manifolds: The convergence case for standard and multiplicative versions, Int. Math. Res. Not., 9:453–486, 2001.

    Article  MathSciNet  Google Scholar 

  12. V.I. Bernik and E.I. Kovalevskaya, Simultaneous inhomogeneous Diophantine approximation of the values of integral polynomials with respect to Archimedean and non-Archimedean valuations, Acta Math. Univ. Ostrav., 14(1):37–42, 2006.

    MathSciNet  MATH  Google Scholar 

  13. N. Budarina and D. Dickinson, p-adic Diophantine approximation on the Veronese curve with a nonmonotonic error, Tr. Inst. Mat., Minsk, 15(1):98–104, 2007.

    Google Scholar 

  14. N. Budarina and D. Dickinson, Diophantine approximation on nondegenerate curves with nonmonotonic error function, Bull. London Math. Soc., 41(1):137–146, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  15. N. Budarina and E. Zorin, Nonhomogeneous analogue of Khintchine theorem for divergence case for simultaneous approximations in the different metrics, Šiauliai Math. Semin., 4(12):21–33, 2009.

    MathSciNet  Google Scholar 

  16. A. Khintchine, Einige Sätze über Kettenbrüche mit Anwendungen auf die Theorie der diophantischen Approximationen, Math. Ann., 92:115–125, 1924.

    Article  MathSciNet  MATH  Google Scholar 

  17. D.Y. Kleinbock and G.A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. Math., 148:339–360, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  18. D. Kleinbock and G. Tomanov, Flows on S-arithmetic homogeneous spaces and applications to metric Diophantine approximation, Comment. Math. Helv., 82:519–581, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  19. E. Kovalevskaya, On the exact order of approximation to zero by values of integral polynomials in Q p , preprint No. 8(547), Institute Mathematics National Academy Sciences Belarus, Minsk, 1998.

    Google Scholar 

  20. J. Kubilius, On an application of I.M. Vinogradov’s method to the solution of a problem of the metrical theory of numbers, Dokl. Akad. Nauk SSSR, 67:783–786, 1949.

    MATH  Google Scholar 

  21. K. Mahler, Über das Mass der Menge aller S-Zahlen, Math. Ann., 106:131–139, 1932.

    Article  MathSciNet  Google Scholar 

  22. A. Mohammadi and A. Salehi Golsefidy, S-arithmetic Khintchine-type theorem, Geom. Funct. Anal., 19:1147–1170, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Mohammadi and A. Salehi Golsefidy, Simultaneous Diophantine approximation in nondegenerate p-adic manifolds, Isr. J. Math. (forthcoming).

  24. V. Sprindžuk, Mahler’s Problem in the Metric Theory of Numbers, Transl. Math. Monogr., Vol. 25, Amer. Math. Soc., Providence, RI, 1969.

    Google Scholar 

  25. F. Želudevich, Simultane diophantische Approximationen abhängiger Grössen in mehreren Metriken, Acta Arith., 46:285–296, 1986.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Khabarovsk Division of Institute for Applied Mathematics, Far Eastern Branch of the Russian Academy of Science, Dzerzhinskogo 54, Khabarovsk, Russia

    Natalia Budarina

Authors
  1. Natalia Budarina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Natalia Budarina.

Additional information

*Supported by the Science Foundation Ireland, grant SFI-RFP08/MTH1512.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Budarina, N. Simultaneous diophantine approximation in the real and p-adic fields with nonmonotonic error function* . Lith Math J 51, 461–471 (2011). https://doi.org/10.1007/s10986-011-9140-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10986-011-9140-6

Keywords

MSC

Search

Navigation