Small discs containing conjugate algebraic integers

Abstract

In this note we show that, for any ξ ∈ R, there is an infinite set of positive integers S such that, for each d ∈ S, the open disc with center at ξ and radius 1 + (log log d) 2 /(2 log d) contains a full set of conjugates of an algebraic integer of degree d. A slightly better bound on the radius is established when ξ ∈ Q \ Z.