We give an approximation inequality for closed subschemes on surfaces, generalizing works of Ru-Vojta, Ru-Wang, Vojta, and Heier-Levin simultaneously. The new method estimates beta constant of closed subschemes with respect to certain divisors. As applications, we bound the greatest common divisors evaluated at integral points on surfaces, and from this we show the degeneracy of certain integral points. We also prove results on the degeneracy of solutions of certain type of S-unit equations. (joint work with Levin and Xiao)
