  A long-standing conjecture of Littlewood about simultaneous Diophantine
approximation has an analogous problem for a field of formal Laurent series
$\mathbb{F}(\!(t^{-1})\!)$. That is, we can ask whether for any series
$\Theta$, $\Phi$ and any $\epsilon>0$, there is a polynomial $\alpha$ such that
$|\alpha|\langle\alpha\Theta\rangle\langle\alpha\Phi\rangle$ where
$\langle\Theta\rangle=\underset{\beta\in\mathbb{F}[t]}{\inf}|\Theta-\beta|$. 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.
