  In this paper, we establish asymptotic formulae with optimal errors for the
number of rational points that are close to a planar curve, which unify and
extend the results of Beresnevich-Dickinson-Velani and Vaughan-Velani.
Furthermore, we complete the Lebesgue theory of Diophantine approximation on
weakly non-degenerate planar curves that was initially developed by
Beresnevich-Zorin in the divergence case.
