Van Lambalgen's theorem and high degrees (Q540405)

In the paper under review, the authors investigate when and where van Lambalgen's theorem fails respectively holds. They prove that: {\parindent5.5mm \begin{itemize}\item[1)] Each high degree contains a recursively random real so that van Lambalgen's theorem fails; \item[2)] Each hyperimmune degree contains a Kurtz random real so that van Lambalgen's theorem fails; \item[3)] There is a recursively random real that is not Martin-Löf random and satisfies van Lambalgen's theorem for recursive, Schnorr and Kurtz randomness. \end{itemize}}