Given two polynomials $P,Q$ and consider $D=\{\tfrac{P(n)}{Q(n)}:n\in\mathbb{N}\}$, can one find polynomials $P^\prime,Q^\prime$ such that $D-D=\{\tfrac{P(n)}{Q(n)}-\tfrac{P(m)}{Q(m)}:n,m\in\mathbb{N}\}=\{\tfrac{P^\prime(N)}{Q^\prime(N)}:N\in\mathbb{N}\}$?
Clearly, this can happen. For example, one can check that with $D=\{n^2-n+1:n\in\mathbb{N}\}$, we have $D-D=2\mathbb{Z}$. (so one would have to change $\mathbb{N}$ for $\mathbb{Z}$ but that would also be okay)