Abstract. We study algebraic properties of cost functions. We give an application: building sets close to being Turing complete. 1.

Abstract. Let A be c.e. Then A is strongly jump traceable if and only if A is Turing below each superhigh Martin-Löf-random set. 1.

Abstract. Demuth tests generalize Martin-Löf tests (Gm)m∈N in that one can exchange the m-th component for a computably bounded number of times. A set Z ⊆ N fails a Demuth test if Z is in infinitely many final versions of the Gm. If we only allow Demuth tests such that Gm ⊇ Gm+1 for each m, we have weak Demuth randomness. We show that a weakly Demuth random set can be high, 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 Π 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. 1.

We survey results relating the computability and randomness aspects of sets of natural numbers. Each aspect corresponds to several mathematical properties. Properties originally defined in very different ways are shown to coincide. For instance, lowness for ML-randomness is equivalent to K-triviality. We include some interactions of randomness with computable analysis. Mathematics Subject Classification (2010). 03D15, 03D32. Keywords. Algorithmic randomness, lowness property, K-triviality, cost function.

1.1. Original result 1 1.2. New results 2

Abstract. Consider a Martin-Löf random ∆ 0 2 set Z. We give lower bounds for the number of changes of Zs ↾n for computable approximations of Z. We show that each nonempty Π 0 1 class has a low member Z with a computable approximation that changes only o(2 n) times. We prove that each superlow ML-random set already satisfies a stronger randomness notion called balanced randomness, which implies that for each computable approximation and each constant c, there are infinitely many n such that Zs ↾n changes more than c2 n times. 1

Abstract. We show that every strongly jump-traceable set is K-trivial. Unlike other results, we do not assume that the sets in question are computably enumerable. 1.

1.1. Original result 2 1.2. New results 2

Abstract. Demuth tests generalize Martin-Löf tests (Gm)m∈N in that one can exchange the m-th component a computably bounded number of times. A set Z ⊆ N fails a Demuth test if Z is in infinitely many final versions of the Gm. If we only allow Demuth tests such that Gm ⊇ Gm+1 for each m, we have weak Demuth randomness. We show that a weakly Demuth random set can be high and ∆ 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 Π 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. 1.

Abstract. Demuth tests generalize Martin-Löf tests (Gm)m∈N in that one can exchange the m-th component for a computably bounded number of times. A set Z ⊆ N fails a Demuth test if Z is in infinitely many final versions of the Gm. If we only allow Demuth tests such that Gm ⊇ Gm+1 for each m, we have weak Demuth randomness. We show that a weakly Demuth random set can be high, 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 Π 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. 1.