TITLE: Cohesive set with degree below non-high Martin-Löf random reals
QUESTION [8 upvotes]: A set A is cohesive if $A\subseteq ^* W_e$ or $A\subseteq^* \bar{W_e}$ for each $e\in \omega$ (standard enumeration of r.e. sets). By Jockusch and Stephan's 1993 paper 'A cohesive set which is not high.', this is equivalent to saying that A is r-cohesive, i.e. restricted to those recursive sets $W_e$. My question is if given a Martin-Löf random real R whose degree is not high, is it possible to have $A\leq_T R$ with A being (not high) cohesive?

REPLY [2 votes]: I'm fairly certain I can construct such an $A$ and $R$ using a priority construction which builds them both simultaneously.  It'd use $\Pi^0_2$-guessing about whether or not intersections are infinite to build $A$, and impose restraint to make $R$ non-high.  I haven't read Jockusch and Stephan's construction, but I assume this is roughly how it went.
The added wrinkle is making $R$ random.  To do that, we'd fix a small but sufficiently universal $\Sigma^0_1$-class $U$, and make $R$ run away from $U$.  For example, make $U$ be the 5th element of a universal Martin-Löf test.  Whenever $R$ looks to be in danger of being covered by $U$, we'd choose new measure and move $R$ there.
Unfortunately, it's a fairly involved construction, and there would need to be some calculations to show that there's always space available for $R$ to run to.  So I can't give full details, and I can't promise that it's correct, but I think it is.  Sorry I can't give anything more definite.