Inhomogeneous Diophantine approximation over fields of formal power series

TL;DR

This paper establishes a precise inhomogeneous approximation theorem over fields of formal power series and explores approximation dynamics under group actions in this setting.

Contribution

It provides a sharp analogue of Minkowski's inhomogeneous approximation theorem over fields of power series and analyzes orbit-based approximation in this context.

Findings

Proved a sharp inhomogeneous approximation theorem for fields of power series.

Analyzed approximation of points by $SL_2( ext{polynomials})$-orbits.

Extended classical Diophantine approximation results to non-Archimedean fields.

Abstract

We prove a sharp analogue of Minkowski's inhomogeneous approximation theorem over fields of power series $\mathbb{F}_q((T^{-1}))$. Furthermore, we study the approximation to a given point $\underline{y}$ in $\mathbb{F}_q((T^{-1}))^2$ by the $SL_2(\mathbb{F}_q[T])$-orbit of a given point $\underline{x}$ in $\mathbb{F}_q((T^{-1}))^2$.