On well-approximable matrices over a field of formal series

Abstract

It is a well-known result in Diophantine approximation over the reals that the Lebesgue measure of the elements $x \in {\mathbb{R}}$, such that $\Vert qx \Vert < {1}/{q^v}$ for infinitely many $q \in {\mathbb{R}}$, is full for $v \leq 1$ and null otherwise, where $\Vert\cdot\Vert$ denotes the distance to nearest integer. Also, the Hausdorff dimension of the set of these numbers is known to be ${2}({v\,{+}\,1})$ for $v > 1$. Similar results are known for matrices over the reals as well as the $p$-adics. In this paper, we prove the corresponding multi-dimensional result for matrices over the field of Laurent series with coefficients from a given finite field.