Exact Approximation In The Field Of Formal Series

Abstract

In this article, we determine the Hausdorff dimension of the set of exactly $ψ$-approximable vectors over local fields of positive characteristic. This result is the function field analogue of a recent theorem of Bandi and de Saxcé in the real setting~\cite{bandi2023hausdorff}, and extends the main theorem of Zhang~\cite{MR2834892} to higher dimensions. Our approach adapts the method of Bandi and de Saxcé to the ultrametric setting, enabling us to overcome difficulties arising from the failure of the well-separatedness property for rational functions in higher dimensions.