ON A PROBLEM OF BERNIK, KLEINBOCK AND MARGULIS

BUDARINA, NATALIA

Glasgow mathematical journal, Tome 53 (2011) no. 3, pp. 669-681

Résumé

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.

Détail

DOI : 10.1017/S0017089511000255

Mots-clés : 11J83, 11K60, 11J13

Bibliographie

[1] 1.Badziahin, D., Beresnevich, V. and Velani, S., Inhomogeneous theory of dual Diophantine approximation on manifolds, Preprint, (2010), pp. 1–37. Google Scholar | arXiv

[2] 2.Baker, A., Transcendental number theory (Cambridge University Press, Cambridge, UK, 1975). Google Scholar | DOI

[3] 3.Beresnevich, V., On approximation of real numbers by real algebraic numbers, Acta Arith. 90 (1999), 97–112. Google Scholar | DOI

[4] 4.Beresnevich, V., A Groshev type theorem for convergence on manifolds, Acta Math. Hungar. 94 (2002), 99–130. Google Scholar | DOI

[5] 5.Beresnevich, V., On a theorem of V. Bernik in the metric theory of Diophantine approximation, Acta Arith. 117 (1) (2005), 71–80. Google Scholar | DOI

[6] 6.Beresnevich, V., Rational points near manifolds and metric Diophantine approximation, Preprint, (2009), pp. 1–46. Google Scholar | arXiv

[7] 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. Google Scholar | DOI

[8] 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] 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. Google Scholar | DOI

[10] 10.Beresnevich, V., Bernik, V. and Götze, F., The distribution of close conjugate algebraic numbers, Compositio Math. 146 (2010), 1165–1179. Google Scholar | DOI

[11] 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. Google Scholar | DOI

[12] 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. Google Scholar | DOI

[13] 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] 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. Google Scholar | DOI

[15] 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. Google Scholar | DOI

[16] 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] 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] 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. Google Scholar | DOI

[19] 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] 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] 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. Google Scholar | DOI

[22] 22.Bugeaud, Y., Approximation by algebraic integers and Hausdorff dimension, J. Lond. Math. Soc. 65 (2002), 547–559. Google Scholar | DOI

[23] 23.Khintchine, A. J., Einige Satze uber Kettenbruche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann. 92 (1924), 115–125. Google Scholar | DOI

[24] 24.Kleinbock, D. Y. and Margulis, G. A., Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. Math. 148 (2) (1998), 339–360. Google Scholar | DOI

[25] 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] 26.Vaughan, R. C. and Velani, S., Diophantine approximation on planar curves: the convergence theory, Invent. Math. 166 (2006), 103–124. Google Scholar | DOI