  We prove a Diophantine approximation inequality for closed subschemes on
surfaces which can be viewed as a joint generalization of recent inequalities
of Ru-Vojta and Heier-Levin in this context. As applications, we study various
Diophantine problems on affine surfaces given as the complement of three
numerically parallel ample projective curves: inequalities involving greatest
common divisors, degeneracy of integral points, and related Diophantine
equations including families of S-unit equations. We state analogous results in
the complex analytic setting, where our main result is an inequality of Second
Main Theorem type for surfaces, with applications to the study and value
distribution theory of holomorphic curves in surfaces.
