The distribution of close conjugate algebraic numbers

TL;DR

This paper studies the distribution of real algebraic numbers with close conjugates, providing new bounds and estimates that improve upon previous results, using a novel approach involving derivatives of polynomials.

Contribution

It introduces a new method for analyzing algebraic numbers with close conjugates by implicitly tailoring polynomials through their derivatives, leading to sharper bounds.

Findings

Establishes the ubiquity of algebraic numbers with close conjugates in the real line.

Provides sharp quantitative bounds on their number.

Improves bounds on the minimal distance between conjugate algebraic numbers.

Abstract

We investigate the distribution of real algebraic numbers of a fixed degree having a close conjugate number, 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 of Bugeaud and Mignotte. So far the results a la Bugeaud and Mignotte 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. The applications of our main theorem considered in this paper include generalisations of a theorem of Baker and Schmidt and a theorem of Bernik, Kleinbock and Margulis in the metric theory of Diophantine approximation.