  For a set $A \subset \mathbb{N}$ we characterize in terms of its density when
there exists an infinite set $B \subset \mathbb{N}$ and $t \in \{0,1\}$ such
that $B+B \subset A-t$, where $B+B : =\{b_1+b_2\colon b_1,b_2 \in B\}$.
Specifically, when the lower density $\underline{d}(A) >1/2$ or the upper
density $\overline{d}(A)> 3/4$, the existence of such a set $B\subset
\mathbb{N}$ and $t\in \{0,1\}$ is assured. Furthermore, whenever
$\underline{d}(A) > 3/4$ or $\overline{d}(A)>5/6$, we show that the shift $t$
is unnecessary and we also provide examples to show that these bounds are
sharp. Finally, we construct a syndetic three-coloring of the natural numbers
that does not contain a monochromatic $B+B+t$ for any infinite set $B \subset
\mathbb{N}$ and number $t \in \mathbb{N}$.
