On the Littlewood conjecture in fields of power series

Abstract

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