On the Littlewood conjecture in fields of power series

Let $\text{k}$ be an arbitrary field. For any fixed badly approximable power series $\mathrm{\Theta }$ in $\text{k}((X^{-1}))$, we give an explicit construction of continuum many badly approximable power series $\mathrm{\Phi }$ for which the pair $(\mathrm{\Theta },\mathrm{\Phi })$ satisfies the Littlewood conjecture. We further discuss the Littlewood conjecture for pairs of algebraic power series.