On Hausdorff dimension in inhomogeneous Diophantine approximation over global function fields

Abstract

We study inhomogeneous Diophantine approximation over the completion Kv of a function field K (over a finite field) for a discrete valuation v, with affine algebra Rv. We obtain an effective upper bound for the Hausdorff dimension of the set BadA(ϵ)={θ∈Kvm:lim inf(p,q)∈Rvm×Rvn,‖q‖→∞lim inf(p,q)∈Rvm×Rvn,‖q‖→∞‖q‖n‖Aq−θ−p‖m⩾ϵ}, of ϵ-badly approximable targets θ∈Kvm given A∈Mm,n(Kv), using an effective version of entropy rigidity in homogeneous dynamics for some diagonal action on the space of Rv-grids. We characterize matrices A for which BadA(ϵ) has full Hausdorff dimension for some ϵ>0 by a Diophantine condition of singularity on average. Our methods work for the approximation using weighted ultrametric distances.