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

TL;DR

This paper explores a combinatorial approach to a version of Littlewood's conjecture within fields of formal Laurent series over finite fields, linking orbit dynamics to Diophantine approximation.

Contribution

It introduces a novel connection between orbit combinatorics under semigroup actions and Diophantine approximation in finite fields, addressing an analogue of Littlewood's conjecture.

Findings

Established a link between orbit combinatorics and Diophantine approximation

Provided insights into the conjecture for finite fields

Extended understanding of Littlewood's conjecture in formal series fields

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 $|\alpha|\langle\alpha\Theta\rangle\langle\alpha\Phi\rangle$ 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.