2015/12/22 by Efrat Bank, Bank, Efrat, Erez Nesharim +3
Mathematics · #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1512.07231
openalex publication_date 2015/12/22 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
This paper deals with the analogue of Inhomogeneous Diophantine Approximation\nin function fields. The inhomogeneous approximation constant of a Laurent\nseries \θ\∈ mathbb Fq\(\(\(1)/(t)\)\) with respect\nto \γ\∈ mathbb Fq\(\(\(1)/(t)\)\) is defined to be\nc(\θ,\γ)=\inf0\≠ N\∈ mathbb Fq\[t\]|N|\⋅|\⟨\nN\θ - \γ \⟩|. We show that for every \θ there exists\n\γ such that c(\θ,\γ)\≥ q-2, and find a sufficient\ncondition on \θ which forces c(\θ,\γ) \≤ q-2 for every\n\γ. Given \θ, we prove that the set\nBA\θ= \\γ\∈ mathbb\nFq\(\(\(1)/(t)\)\) ;: ; c(\θ,\γ)>0 \ has\nfull Hausdorff dimension. Our methods allow us to solve the case of vectors in\n mathbb Fq\(\(\(1)/(t)\)\)d as well. Our results offer\na strengthening to analogues of results for real inhomogeneous approximation.\n