Projects per year
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.
Original language  English 

Pages (fromto)  11651179 
Number of pages  15 
Journal  Compositio Mathematica 
Volume  146 
Issue number  5 
Early online date  23 Jun 2009 
DOIs  
Publication status  Published  Sept 2010 
Keywords
 polynomial root separation
 Diophantine approximation
 approximation by algebraic numbers
 HAUSDORFF DIMENSION
 DIOPHANTINE APPROXIMATION
 POLYNOMIALS
Projects
 2 Finished

Effective methods in metrical Diophantine approx
1/04/08 → 31/03/10
Project: Research project (funded) › Research

Geometrical, dynamical and transference principles in nonlinear Diophantine approximation and applications
1/10/05 → 30/09/10
Project: Research project (funded) › Research