A classical theorem in computability is that every promptly simple set can be cupped in the Turing degrees to some complete set by a low c.e. set. A related question due to A. Nies is whether every promptly simple set can be cupped by a superlow c.e. set, i.e. one whose Turing jump is truth-table reducible to the halting problem ∅'. A negative answer to this question is provided by giving an explicit construction of a promptly simple set that is not superlow cuppable. This problem relates to effective randomness and various lowness notions.
David Diamondstone. "Promptness does not imply superlow cuppability." J. Symbolic Logic 74 (4) 1264 - 1272, December 2009. https://doi.org/10.2178/jsl/1254748690