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

TL;DR

This paper investigates inhomogeneous Diophantine approximation over function fields, establishing bounds on approximation constants, conditions for extremal values, and the Hausdorff dimension of related sets, extending classical real number results to function fields.

Contribution

It provides the first results on bounds and conditions for inhomogeneous approximation constants in function fields, including the full Hausdorff dimension of certain approximation sets.

Findings

Existence of $ heta,\, ext{and}\, orall heta, ext{there exists }\, ext{a }\, ext{gamma with } c( heta, ext{gamma}) \\geq q^{-2}

A sufficient condition on $ heta$ for all $ ext{gamma}$ to satisfy } c( heta, ext{gamma}) \\leq q^{-2}

The set of $ ext{gamma}$ with positive approximation constant has full Hausdorff dimension

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.