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

Abstract

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 $|\alpha |⟨\alpha \Theta ⟩⟨\alpha \Phi ⟩$ where $⟨\Theta ⟩=\underset{\beta \in \mathbb{F}[t]}{\inf }|\Theta -\beta |$. 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.

The previous claim about the counterexample was withdrawn

References in corpus (1)

• Equidistribution and counting under equilibrium states in negative curvature and trees. Applications to non-Archimedean Diophantine approximation