2023/07/25 by Mumtaz Hussain, Hussain, Mumtaz, Benjamin Ward +1 · 3 citations
Mathematics · #11J13 #11J71 #11J83 #Advanced Topology and Set Theory #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2307.13210
openalex publication_date 2023/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a multidimensional weighted analogue of the well-known theorem of Kurzweil (1955) in the metric theory of inhomogeneous Diophantine approximation. Let A be matrix of real numbers, Ψ an n-tuple of monotonic decreasing functions, and let WA(Ψ) be the set of points that infinitely often lie in a Ψ(q)-neighbourhood of the sequence \Aq\q∈ℕ. We prove that the set WA(Ψ) has zero-full Lebesgue measure under convergent-divergent sum conditions with some mild assumptions on A and the approximating functions Ψ. We also prove the Hausdorff dimension results for this set. Along with some geometric arguments, the main ingredients are weighted ubiquity and weighted mass transference principle introduced recently by Kleinbock & Wang (Adv. Math. 2023), and Wang & Wu (Math. Ann. 2021) respectively.