  We study sets of the form $A = \big\{ n \in \mathbb N \big| \lVert p(n)
\rVert_{\mathbb R / \mathbb Z} \leq \varepsilon(n) \big\}$ for various real
valued polynomials $p$ and decay rates $\varepsilon$. In particular, we ask
when such sets are bases of finite order for the positive integers. We show
that generically, $A$ is a basis of order $2$ when $\operatorname{deg} p \geq
3$, but not when $\operatorname{deg} p = 2$, although then $A + A$ still has
asymptotic density $1$.
