Explicit bounds for rational points near planar curves and metric\n Diophantine approximation

2010/02/14 by Victor Beresnevich, Beresnevich, Victor, Evgeniy Zorin +1
Mathematics · #11J13 #11J83 #11K60 #Algebraic Geometry and Number Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1002.2803

openalex publication_date 2010/02/14 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

The primary goal of this paper is to complete the theory of metric\nDiophantine approximation initially developed in [Ann. of Math.(2) 166 (2007),\np.367-426] for C³ non-degenerate planar curves. With this goal in mind, here\nfor the first time we obtain fully explicit bounds for the number of rational\npoints near planar curves. Further, introducing a perturbational approach we\nbring the smoothness condition imposed on the curves down to C¹ (lowest\npossible). This way we broaden the notion of non-degeneracy in a natural\ndirection and introduce a new topologically complete class of planar curves to\nthe theory of Diophantine approximation. In summary, our findings improve and\ncomplete the main theorems of [Ann. of Math.(2) 166 (2007), p.367-426] and\nextend the celebrated theorem of Kleinbock and Margulis appeared in [Ann. of\nMath.(2), 148 (1998), p.339-360] in dimension 2 beyond the notion of\nnon-degeneracy.\n

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