Simultaneous diophantine approximation for a restricted class of pairs of real numbers

We prove that the Littlewood conjecture is satisfied for a restricted class of pairs $(\alpha,\beta)$ of badly approximable numbers. We use the localization of the roots of a cubic equation with coefficients depending on the diophantine properties for the considered pair $(\alpha,\beta)$. The estimates of the roots rely on the properties of the denominators of the convergents of the continued fraction expansion of $\alpha$ and $\beta$.