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Reals which Compute Little

We investigate combinatorial lowness properties of sets of natural numbers (reals). The real A is super-low if A′ ≤tt ∅′, and A is jump-traceable if the values of {e}A(e) can be effectively approximated in a sense to be specified. We investigate those properties, in particular showing that super-lowness and jump-traceability coincide within the r.e. sets but none of the properties implies the other within the ω-r.e. sets. Finally we prove that, for any low r.e. set B, there is is a K-trivial set A 6≤T B.