Computably enumerable sets below random sets

We use Demuth randomness to study strong lowness properties of computably enumerable sets, and sometimes of sets.(1) A set A⊆N is called a base for Demuth randomness if some set Y Turing above A is Demuth random relative to A. We show that there is an incomputable, computably enumerable base for Demuth randomness, and that each base for Demuth randomness is strongly jump-traceable.(2) We obtain new proofs that each computably enumerable set below all superlow (superhigh) Martin-Löf random sets is strongly jump traceable, using Demuth tests.(3) The sets Turing below each ω2-computably approximable Martin-Löf random set form a proper subclass of the bases for Demuth randomness, and hence of the strongly jump traceable sets.(4) The c.e. sets Turing below each ω2-computably approximable Martin-Löf random set satisfy a new, very strong combinatorial lowness property called ω-traceability.