Infinite unrestricted sumsets of the form B + B $B+B$ in sets with large density

For a set $A\subset N$, we characterize the existence of an infinite set $B\subset N$ and $t\in \{0,1\}$ such that $B+B\subset A-t$, where $B+B=\{b_1+b_2:b_1,b_2\in B\}$, in terms of the density of the set $A$. Specifically, when the lower density $\underline{d}\left(A\right)>1/2$ or the upper density $\overline{d}\left(A\right)>2/3$, the existence of such a set $B\subset N$ and $t\in \{0,1\}$ is assured. Furthermore, whenever $\underline{d}\left(A\right)>3/4$ or $\overline{d}\left(A\right)>5/6$, 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 $B\subset N$ and number $t\in N$.