Lowness Properties and Approximations of the Jump

We study and compare two combinatorial lowness notions: strong jump-traceability and well-approximability of the jump, by strengthening the notion of jump-traceability and super-lowness for sets of natural numbers. A computable non-decreasing unbounded function h is called an order function. Informally, a set A is strongly jumptraceable if for each order function h, for each input e one may effectively enumerate a set Te of possible values for the jump JA(e), and the number of values enumerated is at most h(e). A′ is well-approximable if can be effectively approximated with less than h(x) changes at input x, for each order function h. We prove that there is a strongly jump-traceable set which is not computable, and that if A′ is well-approximable then A is strongly jump-traceable. For r.e. sets, the converse holds as well. We characterize jump-traceability and the corresponding strong variant in terms of Kolmogorov complexity, and we investigate other properties of these lowness notions.