Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-07T12:31:53.319Z Has data issue: false hasContentIssue false
  • English
  • Français

HOW SMALL CAN POLYNOMIALS BE IN AN INTERVAL OF GIVEN LENGTH?

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  13 March 2019

VASILI BERNIK  and
STEPHEN Mc GUIRE
VASILI BERNIK
Affiliation:
Institute of Mathematics of the National Academy of Sciences of Belarus, Minsk, Belarus e-mail: [email protected]
STEPHEN Mc GUIRE
Affiliation:
Department of Mathematics, National University of Ireland Maynooth, Maynooth, Kildare, Ireland e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we provide two new extensions to a lemma of Bernik (1983). Applications are also discussed.

MSC classification

Primary: 11J83: Metric theory
Secondary: 11K60: Diophantine approximation 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 2 , May 2020 , pp. 261 - 280
Copyright
Copyright © Glasgow Mathematical Journal Trust 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baker, A. and Schmidt, W. M., Diophantine approximation and Hausdorff dimension, Proc. Lond. Math. Soc. 21 (1970), 1–11.CrossRefGoogle Scholar
Baker, R. C., Sprindžuk’s theorem and Hausdorff dimension, Mathematika 23(2) (1976), 184–197.CrossRefGoogle Scholar
Beresnevich, V., On approximation of real numbers by real algebraic numbers, Acta Arith. 90 (1999), 97–112.CrossRefGoogle Scholar
Beresnevich, V., Bernik, V. I., and Götze, F., Integral polynomials with small discriminants and resultants, Adv. Math. 298 (2016), 393–412.10.1016/j.aim.2016.04.022CrossRefGoogle Scholar
Bernik, V. I., A metric theorem on the simultaneous approximation of zero by the values of integral polynomials, Math. USSR-Izv. 16(1) (1981), 21–40.CrossRefGoogle Scholar
Bernik, V. I., Application of the Hausdorff dimension in the theory of Diophantine approximation, Acta Arith. 42(3) (1983), 219–253. (in Russian)Google Scholar
Bernik, V. I., Budarina, N., Dickinson, D., and Mc Guire, S., The distribution of algebraic conjugate points, in preparation.Google Scholar
Bernik, V. I. and Kalosha, N. I., Approximation of zero by integer polynomials in space $\mathbb{R}\times\mathbb{C}\times {\mathbb{Q}}_{p}$, Proc. Nat. Acad. Sci. Belarus Phis. Math. Ser. 1 (2004), 121–123.Google Scholar
Besicovitch, A., Sets of fractional dimensions. IV: On rational approximation to real numbers, J. Lond. Math. Soc.9 (1934), 126–131.CrossRefGoogle Scholar
Budarina, N. and Dickinson, D., Simultaneous Diophantine approximation of integral polynomials in the different metrics, Chebyshevskii Sbornik 9(1) (2008), 169–184.Google Scholar
Feldman, N. I., Approximation of certain transcendental numbers, I. Izv. Akad. Nauk SSSR Ser. Mat. 15 (1951), 53–79; English transl. in Amer. Math. Soc. Transl. 59(2) (1966).Google Scholar
Jarník, V.. Diophantische approximationen und hausdorffsches maß, Rec. Math. Moscou 36 (1929), 371–382.Google Scholar
Kasch, F. and Volkmann, B., Zur Mahlerschen Vermutung über S-Zahlen, Math. Ann. 136 (1958), 442–453. (in German)CrossRefGoogle Scholar
Koleda, D. V., An upper bound for the number of integral polynomials of third degree with a given bound for discriminants, Vestsí Nats. Akad. Navuk Belarusí Ser. Fíz.-Mat. Navuk 3 (2010), 10–16. (in Russian)Google Scholar
Koleda, D. V. and Korlukova, I. A., Asymptotic quantity of integral quadratic polynomials with bounded discriminants, Vesnik of Yanka Kupala State University of Grodno, Series 2. 2(151) (2013), 6–10.Google Scholar
Kovalevskaya, E., A metric theorem on the exact order of approximation of zero by values of integer polynomials in $\mathbb{Q}_p$, Dokl. Nats. Akad. Nauk Belarusi 43(5) (1999), 34–36.Google Scholar
Mahler, K., Über das Mass der Menge aller S-Zahlen, Math. Ann. 106 (1932), 131–139.10.1007/BF01455882CrossRefGoogle Scholar
Pereverzeva, N. A., The distribution of vectors with algebraic coordinates in $\mathbb{R}^2$, Vestsi Akad. Naavuk BSSR. Ser. Fiz.-Mat. Navuk 4 (1987), 114–116, 128. (in Russian)Google Scholar
Schmidt, W. M., Bounds for certain sums; a remark on a conjecture of Mahler, Trans. Amer. Math. Soc. 101 (1961), 200–210.Google Scholar
Sprindžuk, V. G., Mahler’s problem in the metric theory of numbers, vol. 25 (American Mathematical Society, Providence, RI, 1969).Google Scholar
Volkmann, B., Zur metrischen Theorie der S-Zahlen, J. Reine Angew. Math. 209 (1962), 201–210.Google Scholar