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Abstract
The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in Beresnevich et al. (2007) [10] for C3 nondegenerate planar curves. With this goal in mind, here for the first time we obtain fully explicit bounds for the number of rational points near planar curves. Further, introducing a perturbational approach we bring the smoothness condition imposed on the curves down to C1 (lowest possible). This way we broaden the notion of nondegeneracy in a natural direction and introduce a new topologically complete class of planar curves to the theory of Diophantine approximation. In summary, our findings improve and complete the main theorems of Beresnevich et al. (2007) [10] and extend the celebrated theorem of Kleinbock and Margulis (1998) [20] in dimension 2 beyond the notion of nondegeneracy. (C) 2010 Elsevier Inc. All rights reserved.
Original language  English 

Pages (fromto)  30643087 
Number of pages  24 
Journal  Advances in Mathematics 
Volume  225 
Issue number  6 
DOIs  
Publication status  Published  20 Dec 2010 
Keywords
 Metric simultaneous Diophantine approximation
 Rational points near curves
 Khintchine theorem
 Ubiquity
 HAUSDORFF DIMENSION
 THEOREM
 MANIFOLDS
 CONVERGENCE
 CONVEXITY
 SETS
Activities
 1 Academic
Projects
 1 Finished

Geometrical, dynamical and transference principles in nonlinear Diophantine approximation and applications
1/10/05 → 30/09/10
Project: Research project (funded) › Research