Abstract
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.
Citation
Johanna N.Y. Franklin. Frank Stephan. "Van Lambalgen's Theorem and High Degrees." Notre Dame J. Formal Logic 52 (2) 173 - 185, 2011. https://doi.org/10.1215/00294527-1306181
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