A combinatorial approach to the Littlewood conjecture in a field of formal series

A long-standing conjecture of Littlewood about simultaneous Diophantine approximation has an analogous problem for a field of formal Laurent series $\mathbb{F}(\text{\hspace{0.17em}⁣}(t^{-1})\text{\hspace{0.17em}⁣})$. That is, we can ask whether for any series $\mathrm{\Theta }$, $\mathrm{\Phi }$ and any $ϵ>0$, there is a polynomial $\alpha$ such that $\mathrm{\mid }\alpha \mathrm{\mid }⟨\alpha \mathrm{\Theta }⟩⟨\alpha \mathrm{\Phi }⟩$ where $⟨\mathrm{\Theta }⟩=\mathrm{\mid }\mathrm{\Theta }-\beta \mathrm{\mid }$. If the base field $\mathbb{F}$ is infinite, then the answer is negative due to Davenport and Lewis (1963). We give a connection between the combinatorics of an orbit under a semigroup action and Diophantine approximation problem when $\mathbb{F}$ is finite.