Abstract
Demuth tests generalize Martin-Lof tests (G(m))(m epsilon N) in that one can exchange the m-th component a computably bounded number of times. A set Z subset of N fails a Demuth test if Z is in infinitely many final versions of the G(m).
If we only allow Demuth tests such that G(m) superset of G(m+1) for each m, we have weak Demuth randomness. We show that a weakly Demuth random set can be high and Delta(0)(2) yet not superhigh.
Next, any c.e. set Turing below a Demuth random set is strongly jump-traceable. We also prove a basis theorem for non-empty Pi(0)(1) classes P.
It extends the Jockusch-Soare basis theorem that some member of P is computably dominated. We use the result to show that some weakly 2-random set does not compute a 2-fixed point free function.