Solution of Cassels' Problem on a Diophantine Constant over Function Fields

Abstract

This paper deals with the analogue of Inhomogeneous Diophantine Approximation in function fields. The inhomogeneous approximation constant of a Laurent series $\theta \in {\mathbb{F}}_q\left(\left(\frac{1}{t}\right)\right)$ with respect to $\gamma \in {\mathbb{F}}_q\left(\left(\frac{1}{t}\right)\right)$ is defined to be $c(\theta ,\gamma )={\inf }_{0\ne N\in {\mathbb{F}}_q\left[t\right]}|N|\cdot |⟨N\theta -\gamma ⟩|$. We show that for every $\theta$ there exists $\gamma$ such that $c(\theta ,\gamma )\ge q^{-2}$, and find a sufficient condition on $\theta$ which forces $c(\theta ,\gamma )\le q^{-2}$ for every $\gamma$. Given $\theta$, we prove that the set $B{A}_{\theta }=\left\{\gamma \in {\mathbb{F}}_q\left(\left(\frac{1}{t}\right)\right):c(\theta ,\gamma )>0\right\}$ has full Hausdorff dimension. Our methods allow us to solve the case of vectors in ${\mathbb{F}}_q{\left(\left(\frac{1}{t}\right)\right)}^d$ as well. Our results offer a strengthening to analogues of results for real inhomogeneous approximation.

arXiv commentpublished version