Beresnevich, V., Dickinson, D., Velani, S.L. and Vaughan, R.C. (2007) Diophantine approximation on planar curves and the distribution of rational points (with an appendix "Sum of two squares near perfect squares" by R.C. Vaughan). Annals of Mathematics, 166 (2). pp. 367-426. ISSN 0003-486X
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.
|Institution:||The University of York|
|Academic Units:||The University of York > Mathematics (York)|
|Depositing User:||York RAE Import|
|Date Deposited:||18 Sep 2009 09:34|
|Last Modified:||18 Sep 2009 09:34|