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• W1997555168 abstract "The set A is low for (Martin-Löf) randomness if each random set is already random relative to A. A is K-trivial if the prefix complexity K of each initial segment of A is minimal, namely ∀nK(A↾n)⩽K(n)+O(1). We show that these classes coincide. This answers a question of Ambos-Spies and Kučera in: P. Cholak, S. Lempp, M. Lerman, R. Shore, (Eds.), Computability Theory and Its Applications: Current Trends and Open Problems, American Mathematical Society, Providence, RI, 2000: each low for Martin-Löf random set is Δ20. Our class induces a natural intermediate Σ30 ideal in the r.e. Turing degrees, which generates the whole class under downward closure. Answering a further question in P. Cholak, S. Lempp, M. Lerman, R. Shore, (Eds.), Computability Theory and Its Applications: Current Trends and Open Problems, American Mathematical Society, Providence, RI, 2000, we prove that each low for computably random set is computable." @default.
• W1997555168 created "2016-06-24" @default.
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• W1997555168 date "2005-10-01" @default.
• W1997555168 modified "2023-10-17" @default.
• W1997555168 title "Lowness properties and randomness" @default.
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• W1997555168 doi "https://doi.org/10.1016/j.aim.2004.10.006" @default.
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