On Hausdorff dimension in inhomogeneous Diophantine approximation over global function fields

TL;DR

This paper investigates the Hausdorff dimension of sets of badly approximable points in inhomogeneous Diophantine approximation over global function fields, providing effective bounds and characterizations based on entropy rigidity and Diophantine conditions.

Contribution

It introduces an effective upper bound for the Hausdorff dimension of badly approximable sets over function fields and characterizes matrices with full dimension via a Diophantine singularity condition.

Findings

Derived an effective Hausdorff dimension bound for inhomogeneous approximation sets.

Characterized matrices with full Hausdorff dimension using a Diophantine singularity condition.

Extended methods to weighted ultrametric distances in the approximation context.

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.