On Wirsing's problem in small exact degree

Abstract

We investigate a variant of Wirsing's problem on approximation to a real number by real algebraic numbers of degree exactly n. This has been studied by Bugeaud and Teulie. We improve their bounds for degrees up to n = 7. Moreover, we obtain results regarding small values of polynomials and approximation to a real number by algebraic integers in small prescribed degree. The main ingredient are irreducibility criteria for integral linear combinations of coprime integer polynomials. Moreover, for cubic polynomials these criteria improve results of Győry on a problem of Szegedy.

Optimality of two inequalities for exponents of Diophantine approximation

“…More generally combining (29) and ( 15) enables us to estimate from above (20) for λ ≥ 0, complementary to Corollary 3.2 where λ = n − 1. For large λ, only small improvements compared to the bounds obtained via trivially estimating w * n (ξ) ≥ 1 in place of (28) and using (15) are achieved.…”