On a theorem of Wirsing in Diophantine approximation

Abstract:

Let $n$ and $d$ be integers with $1 \le d \le n-1$. Let $\xi$ be a real number which is not algebraic of degree at most $n$. We establish that there exist an effectively computable constant $c$, depending only on $\xi$ and on $n$, an integer $k$ with $1 \le k \le d$, and infinitely many integer polynomials $P(X)$ of degree $m$ at most equal to $n$ whose roots $\alpha _1, \ldots , \alpha _m$ can be numbered in such a way that \[ |(\xi - \alpha _1) \ldots (\xi - \alpha _k)| \le c H(P)^{-{d \over d+1}n - {1 \over d+1} - 1}. \] This extends a well-known result of Wirsing who dealt with the case $d=1$.