2023/08/31 by Gerardo González Robert, Robert, Gerardo González, Mumtaz Hussain +5
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Number Theory (math.NT) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2308.16603
openalex publication_date 2023/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Theorems of Khintchine, Groshev, Jarník, and Besicovitch in Diophantine approximation are fundamental results on the metric properties of Ψ-well approximable sets. These foundational results have since been generalised to the framework of weighted Diophantine approximation for systems of real linear forms (matrices). In this article, we prove analogues of these weighted results in a range of settings including the p-adics (Theorems 7 and 8), complex numbers (Theorems 9 and 10), quaternions (Theorems 11 and 12), and formal power series (Theorems 13 and 14). The key tools in proving the main parts of these results are the weighted ubiquitous systems and weighted mass transference principle introduced recently by Kleinbock--Wang [Adv. Math. (2023)] and Wang--Wu [Math. Ann. (2021)].