Van Lambalgen's Theorem and High Degrees

Notre Dame Journal of Formal Logic 52 (2):173-185 (2010)
We show that van Lambalgen's Theorem fails with respect to recursive randomness and Schnorr randomness for some real in every high degree and provide a full characterization of the Turing degrees for which van Lambalgen's Theorem can fail with respect to Kurtz randomness. However, we also show that there is a recursively random real that is not Martin-Löf random for which van Lambalgen's Theorem holds with respect to recursive randomness.
Keywords van Lambalgen's Theorem   Schnorr randomness   recursive randomness   Kurtz randomness
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DOI 10.1215/00294527-1306181
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