Best Possible Densities of Dickson m-Tuples, as a Consequence of Zhang–Maynard–Tao

We determine for what proportion of integers h one now knows that there are infinitely many prime pairs p, p + h as a consequence of the Zhang–Maynard–Tao theorem. We consider the natural generalization of this to k-tuples of integers, and we determine the limit of what can be deduced assuming only the Zhang–Maynard–Tao theorem.