Notes on sacks' splitting theorem

We explore the complexity of Sacks' Splitting Theorem in terms of the mind change functions associated with the members of the splits. We prove that, for any c.e. set A, there are low computably enumerable sets A0 & sqcup;A1=A splitting A with A0 and A1 both totally omega 2-c.a. in terms of the Downey-Greenberg hierarchy, and this result cannot be improved to totally omega-c.a. as shown in [9]. We also show that if cone avoidance is added then there is no level below epsilon 0 which can be used to characterize the complexity of A1 and A2.