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    Diophantine approximation on planar curves and the distribution of rational points


    Beresnevich, Victor and Dickinson, Detta and Velani, Sanju and Vaughan, V.C. (2007) Diophantine approximation on planar curves and the distribution of rational points. Annals of Mathematics, 166 (2). pp. 367-426. ISSN 1939-8980

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    Abstract

    Let C be a nondegenerate planar curve and for a real, positive decreasing function ψ let C(ψ) denote the set of simultaneously ψ-approximable points lying on C. We show that C is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on C of C(ψ) is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result. In the case that C is a rational quadric the convergence counterparts of the divergent results are also obtained. Furthermore, for functions ψ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of C(ψ). These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.

    Item Type: Article
    Keywords: Diophantine approximation; planar curves; distribution; rational points;
    Academic Unit: Faculty of Science and Engineering > Mathematics and Statistics
    Item ID: 6920
    Identification Number: https://doi.org/10.4007/annals.2007.166.367
    Depositing User: Dr. Detta Dickinson
    Date Deposited: 25 Jan 2016 09:28
    Journal or Publication Title: Annals of Mathematics
    Publisher: Mathematical Sciences Publishers
    Refereed: Yes
    Funders: INTAS Project 00-429, EPSRC grant GR/R90727/01, NSA grant MDA904-03-1-0082
    URI:

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