The typical Turing degree (Q2874662)

The paper continues the programme initiated by Yates, Martin, Jockusch and Kurtz of understanding the behaviour of ``most'' Turing degrees in terms of both measure and category. The intuition put forward is that many results which hold of 2-random degrees also hold of all predecessors of 2-random degrees.NEWLINENEWLINE The main results proved are:{\parindent=0.6cm\begin{itemize}\item[1.] Every degree bounded by a 2-random degree has a strong minimal cover.\item[2.] Every degree bounded by a 2-random degree satisfies the join property.\item[3.] Every degree bounded by a 2-random degree is the join of two 1-generic degrees.NEWLINENEWLINE\end{itemize}} The proofs are an elaboration on Paris's ``measure risking'' technique. The common features of these proofs are discussed in a methodological way. We remark that recently \textit{L. Bienvenu} and \textit{L. Patey} [``Diagonally non-computable functions and fireworks'', Inf. Comput. (to appear), \url{arXiv:1411.6846}] gave another general presentation of the measure-risking method.