On Hausdorff dimension in inhomogeneous Diophantine approximation over global function fields (2112.04144v3)

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(\epsilon)=\left{\boldsymbol{\theta}\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}-\boldsymbol{\theta}-\mathbf{p}|^m \geq \epsilon \right}, ] of $\epsilon$-badly approximable targets $\boldsymbol{\theta}\in K_v^{\,m}$ 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 $\mathbf{Bad}_A(\epsilon)$ has full Hausdorff dimension for some $\epsilon>0$ by a Diophantine condition of singularity on average. Our methods also work for the approximation using weighted ultrametric distances.