Hausdorff dimension of singular vectors in function fields

TL;DR

This paper calculates the Hausdorff dimension of singular vectors in function fields and provides bounds for the dimension of epsilon-Dirichlet improvable vectors, extending previous Euclidean space results to function fields.

Contribution

It introduces the first computation of Hausdorff dimension for singular vectors in the context of function fields, adapting classical Diophantine approximation results.

Findings

Hausdorff dimension of singular vectors explicitly computed

Bounds established for epsilon-Dirichlet improvable vectors

Extension of Euclidean Diophantine approximation results to function fields

Abstract

We compute the Hausdorff dimension of the set of singular vectors in function fields and bound the Hausdorff dimension of the set of $\varepsilon$-Dirichlet improvable vectors in this setting. This is a function field analogue of the results of Cheung and Chevallier [Duke Math. J. 165 (2016), 2273--2329].