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. 
Andrew Granville, Daniel M. Kane, Dimitris Koukoulopoulos and Robert   J. Lemke Oliver
math.NT math.CO