The two-dimensional diophantine approximation constant. II

Given real numbers α and β, let c₁(α,β) denote the Diophantine approximation constant for the linear form x + αy + βz and let c₂(α, β) denote the corresponding dual constant for the simultaneous approximation of α and β. The paper gives various results about these constants in the case where α and β lie in some real cubic field. For example, it is shown that the suprema of c₁(α, β) and c2(α, β), taken over all α, β such that 1, α, β is an integral basis for a real cubic field, are equal, and a necessary and sufficient condition for this common value to be equal to 2/7 is given.