Hilbert's Tenth Problem and nonrecursive Diophantine sets
Looking at the paper, I believe that what Shlapentokh is saying is that "there is an undecidable Diophantine subset of $\mathbb Z$" is an immediate consequence of "there is an r.e. set that is not recursive" and the MRDP theorem, where by the "MRDP theorem" is meant "A set is Diophantine if and only if it is r.e."  Nothing deeper than that.