On Hausdorff dimension in inhomogeneous Diophantine approximation over global function fields

Abstract

In this paper, we study inhomogeneous Diophantine approximation over the completion $K_v$ of a global function field $K$ (over a finite field) for a discrete valuation $v$, with affine algebra $R_v$. We obtain an effective upper bound for the Hausdorff dimension of the set $$\mathbf{Bad}_A(ε)=\left\{\boldsymbolθ\in K_v^{\,m} : \liminf_{(\mathbf{p},\mathbf{q})\in R_v^{\,m} \times R_v^{\,n}, \|\mathbf{q}\|\to \infty} \|\mathbf{q}\|^n \|A\mathbf{q}-\boldsymbolθ-\mathbf{p}\|^m \geq ε\right\},$$ of $ε$-badly approximable targets $\boldsymbolθ\in K_v^{\,m}$ for a fixed matrix $A\in\mathscr{M}_{m,n}(K_v)$, using an effective version of entropy rigidity in homogeneous dynamics for an appropriate diagonal action on the space of $R_v$-grids. We further characterize matrices $A$ for which $\mathbf{Bad}_A(ε)$ has full Hausdorff dimension for some $ε>0$ by a Diophantine condition of singularity on average. Our methods also work for the approximation using weighted ultrametric distances.