Approximation of complex algebraic numbers by algebraic numbers of bounded degree

Abstract

has infinitely many solutions in algebraic numbers α of degree at most n. Here H(α) denotes the height of α, that is the maximum of the absolute values of the coefficients of the minimal polynomial of α. In 1965, Sprindzhuk proved that kn(ξ) = n + 1 for almost all ξ ∈ R (with respect to the Lebesgue measure on R) and kn(ξ) = n+1 2 for almost all ξ ∈ C (with respect to the Lebesgue measure in C). In 1971 Schmidt proved that if ξ is a real algebraic number of degree d, then kn(ξ) = min(d, n+ 1). So as for approximation by algebraic numbers of degree at most n, real algebraic numbers ξ of degree larger than n show the same behavious as almost all real numbers. Up to now, nobody had computed kn(ξ) for complex algebraic numbers ξ. I will present some new results in this direction, obtained jointly with Yann Bugeaud. These results show that if ξ is complex algebraic of degree d then kn(ξ) = d2 if d ≤ n and kn(ξ) ∈ { n+1 2 , n+2 2 } if d > n. The hard core of the proof of this result is a central result in Diophantine approximation, Schmidt’s Subspace Theorem.