Complexity classes and sparse oracles

The authors obtain positive relativization results. In particular, the goal is to prove statements of the kind: `Given two complexity classes C₁ and C₂, C ₁=C₂ if and only if for every sparse set S, C₁^S=C₂ ^S.´ The authors derive a sufficient condition to obtain such results and, as an application, they prove a general theorem from which, as far as they know, all of the results obtained previously along with new ones can be immediately derived