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The distribution of close conjugate algebraic numbers

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Publication details

JournalCompositio Mathematica
DateE-pub ahead of print - 23 Jun 2009
DatePublished (current) - Sep 2010
Issue number5
Volume146
Number of pages15
Pages (from-to)1165-1179
Early online date23/06/09
Original languageEnglish

Abstract

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 a 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.

    Research areas

  • polynomial root separation, Diophantine approximation, approximation by algebraic numbers, HAUSDORFF DIMENSION, DIOPHANTINE APPROXIMATION, POLYNOMIALS

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