Jump Restricted Interpolation in the Recursively Enumerable Degrees

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Sacks has shown [3, Theorem 3 of ? 6] that if b is r.e., f< 0' < u, f 4b and u is r.e. in 0' then there is an r.e. degree d such that b < d, f ;f d, and di = u. He also showed [3, Theorem 1 of ? 7] that if b, c are r.e. and b < c then there is an r.e. degree d with b < d < c and d' = c'. The methods which Sacks used to prove these theorems and, later, the Density Theorem [4] were combined by the author in [1]. There it is shown that if b, c are r.e. and b < c then there is an r.e. degree d such that b < d < c and d' = b'. In [6] Yates showed that if 0 < f < 0' then there is an r.e. degree d such that d I f and d' = 0. Yates based his proof on entirely different methods which earlier [5] had given the Density Theorem as a by-product. These theorems are all implied by the most general result of the present paper, Theorem 3, which states that if b and c are r.e., b ? c, {ff}iN < O' {f}i6N < c, f, S b and c : gi for all i, {g9}leN < u, b' < u, and u is r.e. in c then there is an r.e. degree d such that b < d < c, f, ;f d % gi for all i, and d' = u. Yates [6, page 260] has conjectured that there are degrees f and u such that 0 < f < O' < u, u is r.e. in 0', and there is no r.e. d such that d I f and di = u. In Corollary 3 Yates' conjecture is disproved by combining our main result with a splitting theorem from [2]. Corollary 3 implies that if b and c are r.e., b ? c, b < fi < c for all i < n, bY = O' < u, and u is r.e. in c then there is an r.e. degree d such that b < d < c, d I f, for all i < n, and d' = u.' The reader is assumed to be familiar with [2]. It should be noted in particular that any Greek letter other than X or [ denotes a special recursive function. In addition, the basic Lemmas 1, 2, and 3 are proved by means of observations on the jumps of sets constructed in the proofs of Theorems 1 and 2 and Lemma 6 of [2].