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The distribution of close conjugate algebraic numbers

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

Published online by Cambridge University Press:  22 June 2010

Victor Beresnevich ,
Vasili Bernik  and
Friedrich Götze
Victor Beresnevich
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, UK (email: vb8@york.ac.uk)
Vasili Bernik
Affiliation:
Institute of Mathematics, Surganova 11, Minsk 220072, Belarus (email: bernik@im.bas-net.by)
Friedrich Götze
Affiliation:
Faculty of Mathematics, University of Bielefeld, 33501 Bielefeld, Germany (email: goetze@math.uni-bielefeld.de)
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Abstract

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We investigate the distribution of real algebraic numbers of a fixed degree that have a close conjugate number, with the distance between the conjugate numbers being given as a function of their height. The main result establishes the ubiquity of such algebraic numbers in the real line and implies a sharp quantitative bound on their number. Although the main result is rather general, it implies new estimates on the least possible distance between conjugate algebraic numbers, which improve recent bounds obtained by Bugeaud and Mignotte. So far, the results à la Bugeaud and Mignotte have relied on finding explicit families of polynomials with clusters of roots. Here we suggest a different approach in which irreducible polynomials are implicitly tailored so that their derivatives assume certain values. We consider some applications of our main theorem, including generalisations of a theorem of Baker and Schmidt and a theorem of Bernik, Kleinbock and Margulis in the metric theory of Diophantine approximation.

Keywords

polynomial root separationDiophantine approximationapproximation by algebraic numbers

MSC classification

Secondary: 11J83: Metric theory 11J13: Simultaneous homogeneous approximation, linear forms 11K60: Diophantine approximation 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 5 , September 2010 , pp. 1165 - 1179
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Baker, A. and Schmidt, W. M., Diophantine approximation and Hausdorff dimension, Proc. London Math. Soc. 21 (1970), 1–11.CrossRefGoogle Scholar
[2]Beresnevich, V., On approximation of real numbers by real algebraic numbers, Acta Arith. 90 (1999), 97–112.CrossRefGoogle Scholar
[3]Beresnevich, V., Rational points near manifolds and metric Diophantine approximation, Preprint (2009), arXiv:0904.0474.Google Scholar
[4]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) (Cambridge University Press, Cambridge, 2002), 260–279.CrossRefGoogle Scholar
[5]Beresnevich, V., Dickinson, D. and Velani, S., Measure theoretic laws for lim sup sets, Mem. Amer. Math. Soc. 179 (2006), no. 846.Google Scholar
[6]Beresnevich, V., Dickinson, D. and Velani, S., Diophantine approximation on planar curves and the distribution of rational points, Ann. of Math. (2) 166 (2007), 367–426, with an appendix by R. C. Vaughan.CrossRefGoogle Scholar
[7]Beresnevich, V. and Velani, S., An inhomogeneous transference principle and Diophantine approximation, Preprint (2008), arXiv:0802.1837, Proc. London Math. Soc. (3), to appear, doi:10.1112/plms/pdq002.CrossRefGoogle Scholar
[8]Bernik, V. I., An application of Hausdorff dimension in the theory of Diophantine approximation, Acta Arith. 42 (1983), 219–253 (in Russian); English translation in Amer. Math. Soc. Transl. 140 (1988), 15–44.Google Scholar
[9]Bernik, V., Götze, F. and Kukso, O., Lower bounds for the number of integral polynomials with given order of discriminants, Acta Arith. 133 (2008), 375–390.CrossRefGoogle Scholar
[10]Bernik, V., Götze, F. and Kukso, O., On the divisibility of the discriminant of an integral polynomial by prime powers, Lith. Math. J. 48 (2008), 380–396.CrossRefGoogle Scholar
[11]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. Not. (2001), 453–486.CrossRefGoogle Scholar
[12]Bugeaud, Y., Approximation by algebraic integers and Hausdorff dimension, J. London Math. Soc. 65 (2002), 547–559.CrossRefGoogle Scholar
[13]Bugeaud, Y. and Mignotte, M., On the distance between roots of integer polynomials, Proc. Edinb. Math. Soc. (2) 47 (2004), 553–556.CrossRefGoogle Scholar
[14]Bugeaud, Y. and Mignotte, M., Polynomial root separation, Int. J. Number Theory (2009), to appear.CrossRefGoogle Scholar
[15]Evertse, J.-H., Distances between the conjugates of an algebraic number, Publ. Math. Debrecen 65 (2004), 323–340.CrossRefGoogle Scholar
[16]Kukso, O. S., Optimal regular systems consisting of the roots of polynomials with small discriminants and their applications, Vestsī Nats. Akad. Navuk Belarusī Ser. Fīz.-Mat. Navuk (Proc. Nat. Acad. Sci. Belarus, Ser. Phys.-Math. Sci.) 2 (2007), 41–47.Google Scholar
[17]Mahler, K., Uber̈ das Maß der Menge aller S-Zahlen, Math. Ann. 106 (1932), 131–139.CrossRefGoogle Scholar
[18]Mahler, K., An inequality for the discriminant of a polynomial, Michigan Math. J. 11 (1964), 257–262.CrossRefGoogle Scholar
[19]Mignotte, M., Some useful bounds, in Computer algebra (Springer, Vienna, 1983), 259–263.CrossRefGoogle Scholar
[20]Sprindžuk, V. G., The proof of Mahler’s conjecture on the measure of the set of S-numbers, Izv. Akad. Nauk SSSR, Ser. Mat. 19 (1965), 191–194 (in Russian).Google Scholar
[21]Sprindžuk, V. G., Mahler’s problem in the metric theory of numbers, Translations of Mathematical Monographs, vol. 25 (American Mathematical Society, Providence, RI, 1969).Google Scholar