2021/12/08 by Kim, Taehyeong, Lim, Seonhee, Paulin, Frédéric · 2 citations
#Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2112.04144
In this paper, we study inhomogeneous Diophantine approximation over the completion Kv of a global 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)=\\boldsymbolθ∈ Kv m : \liminf(p,q)∈ Rv m × Rv n, ‖q‖→ ∞ ‖q‖n ‖Aq-\boldsymbolθ-p‖m ≥ ε\, of ε-badly approximable targets \boldsymbolθ∈ Kv m for a fixed matrix A∈\mathscrMm,n(Kv), using an effective version of entropy rigidity in homogeneous dynamics for an appropriate diagonal action on the space of Rv-grids. We further 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 also work for the approximation using weighted ultrametric distances.