On the Littlewood conjecture in fields of power series

Abstract

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