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

TY - JOUR

T1 - Explicit bounds for rational points near planar curves and metric Diophantine approximation

AU - Beresnevich, Victor

AU - Zorin, Evgeniy

PY - 2010/12/20

Y1 - 2010/12/20

N2 - The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in Beresnevich et al. (2007) [10] for C-3 non-degenerate 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 C-1 (lowest possible). This way we broaden the notion of non-degeneracy 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 non-degeneracy. (C) 2010 Elsevier Inc. All rights reserved.

AB - The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in Beresnevich et al. (2007) [10] for C-3 non-degenerate 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 C-1 (lowest possible). This way we broaden the notion of non-degeneracy 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 non-degeneracy. (C) 2010 Elsevier Inc. All rights reserved.

KW - Metric simultaneous Diophantine approximation

KW - Rational points near curves

KW - Khintchine theorem

KW - Ubiquity

KW - HAUSDORFF DIMENSION

KW - THEOREM

KW - MANIFOLDS

KW - CONVERGENCE

KW - CONVEXITY

KW - SETS

UR - http://www.scopus.com/inward/record.url?scp=77957751166&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2010.05.021

DO - 10.1016/j.aim.2010.05.021

M3 - Article

SN - 0001-8708

VL - 225

SP - 3064

EP - 3087

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 6

ER -