Van Lambalgen's Theorem and High Degrees
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.
Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1303995713
Digital Object Identifier: doi:10.1215/00294527-1306181
Zentralblatt MATH identifier: 05903677
Mathematical Reviews number (MathSciNet): MR2794650