The distribution of close conjugate algebraic numbers

TY - JOUR

T1 - The distribution of close conjugate algebraic numbers

AU - Beresnevich, Victor

AU - Bernik, Vasili

AU - Goetze, Friedrich

PY - 2010/9

Y1 - 2010/9

N2 - 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.

AB - 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.

KW - polynomial root separation

KW - Diophantine approximation

KW - approximation by algebraic numbers

KW - HAUSDORFF DIMENSION

KW - DIOPHANTINE APPROXIMATION

KW - POLYNOMIALS

UR - http://www.scopus.com/inward/record.url?scp=77957233439&partnerID=8YFLogxK

U2 - 10.1112/S0010437X10004860

DO - 10.1112/S0010437X10004860

M3 - Article

SN - 0010-437X

VL - 146

SP - 1165

EP - 1179

JO - Compositio Mathematica

JF - Compositio Mathematica

IS - 5

ER -