𝐾-trivial degrees and the jump-traceability hierarchy

Abstract

For every order h h such that ∑ n 1 / h ( n ) \sum _n 1/h(n) is finite, every K K -trivial degree is h h -jump-traceable. This motivated Cholak, Downey and Greenberg to ask whether this traceability property is actually equivalent to K K -triviality, thereby giving the hoped for combinatorial characterisation of lowness for Martin-Löf randomness. We show however that the K K -trivial degrees are properly contained in those that are h h -jump-traceable for every convergent order h h .