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\(\omega\)-change randomness and weak Demuth randomness. (English) Zbl 1353.03046
Summary: We extend our work on difference randomness. Each component of a difference test is a Boolean combination of two r.e. open sets; here we consider tests in which the \(k^{th}\) component is a Boolean combination of \(g(k)\) r.e. open sets for a given recursive function \(g\). We use this method to produce an alternate characterization of weak Demuth randomness in terms of these tests and further show that a real is weakly Demuth random if and only if it is Martin-Löf random and cannot compute a strongly prompt r.e. set. We conclude with a study of related lowness notions and obtain as a corollary that lowness for balanced randomness is equivalent to being recursive.

MSC:
03D32 Algorithmic randomness and dimension
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