2006/04/29 by Dzmitry Badziahin, Badziahin, Dzmitry, Jason Levesley +1
Mathematics · #11J83 (Primary) 11J13 #11K60 (Secondary) #Analytic Number Theory Research #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.NT #msc:11J13 #msc:11J83 #msc:11K60
paper · pdf · doi:10.48550/arxiv.math/0605004
13 pages
arxiv created 2006/04/29 · openalex publication_date 2006/04/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let C be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation with two independent approximation functions; that is if a certain sum converges then the set of all points (x,y) on the curve which satisfy simultaneously the inequalities ‖ q x ‖ < ψ1(q) and ‖ qy ‖ < ψ2(q) infinitely often has induced measure 0. This completes the metric theory for the Lebesgue case. Further, for the cae of multiplicative approximation ‖ qx ‖ ‖ q y ‖ < ψ(q), we establish a Hausdorff measure convergence result for the same class of curves, the first such result for a general class of manifolds in this particular setup.