A reducibility condition for recursiveness

A result due to Jockusch, equating recursiveness of a set to a reducibility condition on its jump, is sharpened. Introduction. Unexplained notation is taken from Rogers [5]. In an unpublished proof in 1970, Carl Jockusch showed that if A' (see [5, p. 110]). We denote the associated set {xl, . . . , x} by J. If a(0, .. . , 0) = 0 we say t is zero-preserving. Results. THEOREM 1. If A is r.e. and HA <btt 0' then A is recursive. PROOF. Let n be the least integer such that HA <btt 0' with norm bounded by n. Let h be a recursive function such that e E HA X the tt-condition h(e) is satisfied by 0', and each h(e) has norm bounded by n. Assume A nonrecursive. Define ifx 0', Wf(e,x) = A 0 otherwise. Note that if x E 0' then f(e,x) E HAX =e E HA. Define Wg(ey) = W U { y}. If y E A then g(e,y) E HA<= e E HA. Fix e. Set Received by the editors August 25, 1975. AMS (MOS) subject classifications (1970). Primary 02F25, 02F30. 1 This work was supported by NRC grant 126-6324-32. Copyright 6) 1977, American Mathematical Society