Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-27T20:46:47.887Z Has data issue: false hasContentIssue false
  • English
  • Français

ON A PROBLEM OF BERNIK, KLEINBOCK AND MARGULIS

Published online by Cambridge University Press:  01 August 2011

NATALIA BUDARINA
NATALIA BUDARINA*
Affiliation:
Department of Mathematics and Statistics, NUI Maynooth, Maynooth, Co. Kildare, Ireland e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, the Khintchine-type theorems of Beresnevich (Acta Arith. 90 (1999), 97) and Bernik (Acta Arith. 53 (1989), 17) for polynomials are generalised to incorporate a natural restriction on derivatives. This represents the first attempt to solve a problem posed by Bernik, Kleinbock and Margulis (Int. Math. Res. Notices2001(9) (2001), 453). More specifically, the main result provides a probabilistic criterion for the solvability of the system of inequalities |P(x)| < Ψ1(H) and |P′(x)| < Ψ2(H) in integral polynomials P of degree ≤ n and height H, where Ψ1 and Ψ2 are fairly general error functions. The proof builds upon Sprindzuk's method of essential and inessential domains and the recent ideas of Beresnevich, Bernik and Götze (Compositio Math. 146 (2010), 1165) concerning the distribution of algebraic numbers.

Keywords

11J8311K6011J13
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 3 , September 2011 , pp. 669 - 681
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Badziahin, D., Beresnevich, V. and Velani, S., Inhomogeneous theory of dual Diophantine approximation on manifolds, Preprint, http://arxiv.org/abs/1009.5638 (2010), pp. 1–37.Google Scholar
2.Baker, A., Transcendental number theory (Cambridge University Press, Cambridge, UK, 1975).CrossRefGoogle Scholar
3.Beresnevich, V., On approximation of real numbers by real algebraic numbers, Acta Arith. 90 (1999), 97–112.CrossRefGoogle Scholar
4.Beresnevich, V., A Groshev type theorem for convergence on manifolds, Acta Math. Hungar. 94 (2002), 99–130.CrossRefGoogle Scholar
5.Beresnevich, V., On a theorem of V. Bernik in the metric theory of Diophantine approximation, Acta Arith. 117 (1) (2005), 71–80.CrossRefGoogle Scholar
6.Beresnevich, V., Rational points near manifolds and metric Diophantine approximation, Preprint, http://arxiv.org/abs/0904.0474 (2009), pp. 1–46.Google Scholar
7.Beresnevich, V., Bernik, V. I. and Dodson, M. M., Regular systems, ubiquity and Diophantine approximation, in A panorama of number theory or the view from Baker's garden (Zürich, 1999) (Wüstholz, G., editor) (Cambridge University Press, Cambridge, UK, 2002), pp. 260–279.CrossRefGoogle Scholar
8.Beresnevich, V., Dickinson, D. and Velani, S., Measure theoretic laws for lim sup sets, Mem. Amer. Math. Soc. 179 (2006), x + 91.Google Scholar
9.Beresnevich, V., Dickinson, D. and Velani, S., Diophantine approximation on planar curves and the distribution of rational points, Ann. Math. 166 (2) (2007), 367–426. With an Appendix II by R. C. Vaughan.CrossRefGoogle Scholar
10.Beresnevich, V., Bernik, V. and Götze, F., The distribution of close conjugate algebraic numbers, Compositio Math. 146 (2010), 1165–1179.CrossRefGoogle Scholar
11.Beresnevich, V. V., Bernik, V. I., Kleinbock, D. Y. and Margulis, G. A., Metric Diophantine approximation: the Khintchine–Groshev theorem for nondegenerate manifolds, Mosc. Math. J. 2 (2002), 203–225.CrossRefGoogle Scholar
12.Beresnevich, V., Bernik, V. I. and Kovalevskaya, E. I., On approximation of p-adic numbers by p-adic algebraic numbers, J. Number Theory 111 (1) (2005), 33–56.CrossRefGoogle Scholar
13.Bernik, V. I., On the exact order of approximation of zero by values of integral polynomials, Acta Arith. 53 (1989), 17–28 (in Russian).Google Scholar
14.Bernik, V., Kukso, O. and Götze, F., Lower bounds for the number of integer polynomials with given order of discriminants, Acta Arithm. 133 (2008), 375–390.CrossRefGoogle Scholar
15.Bernik, V., Budarina, N. and Dickinson, D., A divergent Khintchine theorem in the real, complex, and p-adic fields, Lith. Math. J. 48 (2) (2008), 158–173.CrossRefGoogle Scholar
16.Bernik, V., Budarina, N. and Dickinson, D., Khinchin's theorem and the approximation of zero by values of integer polynomials in different metrics, Dokl. Akad. Nauk 413 (2) (2007), 151–153.Google Scholar
17.Bernik, V., Budarina, N. and Dickinson, D., Simultaneous Diophantine approximation in the real, complex and p-adic fields, Math. Proc. Cam. Phil. Soc. 149 (2) (2010), 217–227.Google Scholar
18.Bernik, V. I., Kleinbock, D. and Margulis, G. A., Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions, Int. Math. Res. Notices 2001 (9) (2001), 453–486.CrossRefGoogle Scholar
19.Bernik, V. and Shamukova, N., Approximation of real numbers by integer algebraic numbers, and the Khinchin theorem, Dokl. Nat. Akad. Nauk Belarusi. 50 (3) (2006), 30–32.Google Scholar
20.Bernik, V. I. and Vasil'ev, D. V., A Khinchin-type theorem for integer-valued polynomials of a complex variable, Proceedings of the Institute of Mathematics, Vol. 3 (Russian), pp. 10–20.Google Scholar
21.Budarina, N. and Dickinson, D., Diophantine approximation on non-degenerate curves with non-monotonic error function, Bull. Lond. Math. Soc. 41 (1) (2009), 137–146.CrossRefGoogle Scholar
22.Bugeaud, Y., Approximation by algebraic integers and Hausdorff dimension, J. Lond. Math. Soc. 65 (2002), 547–559.CrossRefGoogle Scholar
23.Khintchine, A. J., Einige Satze uber Kettenbruche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann. 92 (1924), 115–125.CrossRefGoogle Scholar
24.Kleinbock, D. Y. and Margulis, G. A., Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. Math. 148 (2) (1998), 339–360.CrossRefGoogle Scholar
25.Sprindžuk, V., Mahler's problem in the metric theory of numbers, Translations of Mathematical Monographs, Vol. 25 (American Mathematical Society, Providence, RI, 1969).Google Scholar
26.Vaughan, R. C. and Velani, S., Diophantine approximation on planar curves: the convergence theory, Invent. Math. 166 (2006), 103–124.CrossRefGoogle Scholar