The question can be stated very simply as:
When is a renewal function $U$ not only subadditive ($U(x+y) \leq U(x) + U(y)$), but strictly subadditive (<)? Or, in other words, when is the renewal function additive, i.e. $U(x+y) = U(x) + U(y)$?
For a proof of the subadditive nature of a renewal function I found in Daley:
Question 2: Are there always $x,y$ such that $U(x+y) < U(x) + U(y)$?