Notes on Sacks' splitting theorem - Ambos-Spies, Klaus; Downey, Rod G.; Monath, Martin; Ng, Keng Meng (The journal of symbolic logic, 2023)
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⊔A1=AA0⊔A1=AA_0\sqcup A_1=A splitting A with A0A0A_0 and A1A1A_1 both totally ω2ω2ω ^2-c.a. in terms of the Downey-Greenberg hierarchy, and this result cannot be improved to totally ωωω -c.a. as shown in [9]. We also show that if cone avoidance is added then there is no level below ε0ε0ε _0 which can be used to characterize the complexity of A1A1A_1 and A2A2A_2.