Effective lower bounds for some linear forms

Abstract:

It is proved that if $1, \alpha ,\beta$ are numbers, linearly independent over the rationals, in a real cubic number field, then given any real number $d \geqslant 2$, for any integers ${x_0},{x_1},{x_2}$ such that $|{\text {norm}}({x_0} + \alpha {x_1} + \beta {x_2})| \leqslant d$, there exist effectively computable numbers $c > 0$ and $k > 0$ depending only on $\alpha$ and $\beta$ such that $|{x_1}{x_2}|{(\log |{x_1}{x_2}|)^{k\log d}}|{x_0} + \alpha {x_1} + \beta {x_2}| > c$ holds whenever ${x_1}{x_2} \ne 0$. It would be of much interest to remove the dependence on d in the exponent of $\log |{x_1}{x_2}|$, for then, among other things, one could deduce, for cubic irrationals, a stronger and effective form of Roth’s Theorem.