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

TL;DR

This paper advances metric Diophantine approximation for planar curves by providing explicit bounds on rational points near these curves, lowering smoothness requirements, and broadening the class of curves considered.

Contribution

It offers the first explicit bounds for rational points near planar curves and extends the theory to $C^1$ curves, broadening the scope of non-degeneracy.

Findings

Derived explicit bounds for rational points near planar curves.

Lowered smoothness condition to $C^1$ for non-degeneracy.

Extended Diophantine approximation results to a larger class of curves.

Abstract

The primary goal of this paper is to complete the theory of metric Diophantine approximation initially developed in [Ann. of Math.(2) 166 (2007), p.367-426] 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 [Ann. of Math.(2) 166 (2007), p.367-426] and extend the celebrated theorem of Kleinbock and Margulis appeared in [Ann. of Math.(2), 148 (1998), p.339-360] in dimension 2 beyond the notion of non-degeneracy.