# zbMATH — the first resource for mathematics

$$\omega$$-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
Full Text:
##### References:
 [1] DOI: 10.1090/S0002-9939-2010-10513-0 · Zbl 1214.03029 · doi:10.1090/S0002-9939-2010-10513-0 [2] DOI: 10.1007/978-3-642-13962-8_18 · Zbl 1286.03140 · doi:10.1007/978-3-642-13962-8_18 [3] Algorithmic randomness and complexity (2010) · Zbl 1221.68005 [4] Commentationes Mathematicae Universitatis Carolinae 23 pp 453– (1982) [5] DOI: 10.1016/j.apal.2011.01.004 · Zbl 1223.03026 · doi:10.1016/j.apal.2011.01.004 [6] Recursively enumerable sets and degrees (1987) [7] DOI: 10.1007/s11856-012-0012-5 · Zbl 1279.03065 · doi:10.1007/s11856-012-0012-5 [8] DOI: 10.1090/S0002-9947-1984-0719661-8 · doi:10.1090/S0002-9947-1984-0719661-8 [9] Computability and randomness (2009) · Zbl 1169.03034 [10] Martin-Löf random and PA-complete sets (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.