# On homogeneous and inhomogeneous Diophantine approximation over the fields of formal power series

## TL;DR

This paper extends classical Diophantine approximation results to fields of power series, establishing analogues of Kronecker's theorem and exploring approximation properties with full Hausdorff dimension in the one-dimensional case.

## Contribution

It introduces the first analogues of key Diophantine approximation theorems over fields of power series, including a quantitative transference inequality and conditions for full Hausdorff dimension.

## Key findings

- Established power series analogue of Kronecker's theorem.
- Derived a quantitative transference inequality.
- Identified conditions for full Hausdorff dimension in one dimension.

## Abstract

We prove over fields of power series the analogues of several Diophantine approximation results obtained over the field of real numbers. In particular we establish the power series analogue of Kronecker's theorem for matrices, together with a quantitative form of it, which can also be seen as a transference inequality between uniform approximation and inhomogeneous approximation. Special attention is devoted to the one dimensional case. Namely, we give a necessary and sufficient condition on an irrational power series $\alpha$ which ensures that, for some positive $\eps$, the set $$ \liminf_{Q \in \mathbb{F}_q[z], \,\, \deg Q \to \infty} \| Q \| \cdot |\langle Q \alpha - \theta \rangle| \geq \eps $$ has full Hausdorff dimension.