Solution of Cassels' Problem on a Diophantine Constant over Function Fields
This paper deals with the analogue of Inhomogeneous Diophantine Approximation in function fields. The inhomogeneous approximation constant of a Laurent series $θ\in\mathbb F_q\left(\left(\frac{1}{t}\right)\right)$ with respect to $γ\in\mathbb F_q\left(\left(\frac{1}{t}\right)\right)$ is defined to be $c(θ,γ)=\inf_{0\neq N\in\mathbb F_q\left[t\right]}|N|\cdot|\langle Nθ- γ\rangle|$. We show that for every $θ$ there exists $γ$ such that $c(θ,γ)\geq q^{-2}$, and find a sufficient condition on $θ$ which forces $c(θ,γ) \leq q^{-2}$ for every $γ$. Given $θ$, we prove that the set $BA_θ=\left\{γ\in\mathbb F_q\left(\left(\frac{1}{t}\right)\right)\;:\; c(θ,γ)>0\right\}$ 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.