Linear forms at a basis of an algebraic number field

It was proved by Cassels and Swinnerton-Dyer that the Littlewood conjecture in simultaneous Diophantine approximation holds for any pair of numbers in a cubic field. Later this result was generalized by Peck to a basis (1,α1,…,αn) of a real algebraic number field of degree at least 3. By transference, this result provides some solutions for the dual form of Littlewoodʼs conjecture. Here we find another solutions, and using Bakerʼs estimates for linear forms in logarithms of algebraic numbers, we discuss whether the result is best possible.