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 $c(\theta,\gamma)=\inf_{0\neq N\in\mathbb F_q\left[t\right]}|N|\cdot|\langle N\theta - \gamma \rangle|$. We show that for every $\theta$ there exists $\gamma$ such that $c(\theta,\gamma)\geq q^{-2}$, and find a sufficient condition on $\theta$ which forces $c(\theta,\gamma) \leq q^{-2}$ for every $\gamma$. Given $\theta$, we prove that the set $BA_{\theta}=\left\{\gamma\in\mathbb F_q\left(\left(\frac{1}{t}\right)\right)\;:\; c(\theta,\gamma)>0\right\}$ 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.