10.1112/S0010437X10004860
Beresnevich, Victor
Victor
Beresnevich
Bernik, Vasili
Vasili
Bernik
Götze, Friedrich
Friedrich
Götze
The distribution of close conjugate algebraic numbers
LONDON MATH SOC
2010
2010-12-07T10:47:30Z
2019-05-10T09:30:23Z
journal_article
https://pub.uni-bielefeld.de/record/1930158
https://pub.uni-bielefeld.de/record/1930158.json
0010-437X
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.