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}q} n }Aq ´θ ´p} m ě ǫ * , of ǫ-badly approximable targets θ P K m v for a fixed matrix A P M m,n pK v q, 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 Bad A pǫq has full Hausdorff dimension for some ǫ ą 0 by a Diophantine condition of singularity on average. Our methods also work for the approximation using weighted ultrametric distances. 1 2 Background material for the lower bound On global function fieldsWe refer for instance to [Gos, Ros], as well as [BPP, §14.2], for the content of this section. Let F q be a finite field with q elements, where q is a positive power of a positive prime. Let K be the function field of a geometrically connected smooth projective curve C over F q , or equivalently an extension of F q of transcendence degree 1, in which F q is algebraically closed. We denote by g the genus of C. There is a bijection between the set of closed points of C and the set of normalized discrete valuations v of K, the valuation of a given element f P K being the order of the zero or the opposite of the order of the pole of f at the given closed point. We fix such an element v throughout this paper, and use the notation K v , O v , π v , k v , q v , | ¨| defined in the introduction. We furthermore denote by deg v the degree of v, so that q v " q deg v .We denote by vol v the normalized Haar measure on the locally compact additive group K v such that vol v pO v q " 1. For any positive integer d, let vol d v be the normalized Haar measure on K d v such that vol d v pO d v q " 1. Note that for every g P GL d pK v q we have