Effective lower bounds for some linear forms

It is proved that if 1, β are numbers, linearly independent over the rationals, in a real cubic number field, then given any real number d > 2, for any integers x₀, x₁, x₂ such that norm(x0 + αX₁ + βX₂)l < d, there exist effectively computable numbers c > 0 and k > 0 depending only on α and β such that X₁+X₂(logx₁x₂)^{k log d}x₀+αx₁=βx₂>holds whenever x₁x₂ ≠ 0. It would be of much interest to remove the dependence on d in the exponent of log x₁x₂ for then, among other things, one could deduce, for cubic irrationals, a stronger and effective form of Roth’s Theorem.