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:	Publisher
Use Licence:	This item is available under a Creative Commons Attribution Non Commercial Share Alike Licence (CC BY-NC-SA). Details of this licence are available here

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