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

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 $B+B:=\{b_1+b_2\text{\hspace{0.17em}⁣}:b_1,b_2\in B\}$. Specifically, when the lower density $\underset{‾}{d}(A)>1\mathrm{/}2$ or the upper density $\bar{d}(A)>3\mathrm{/}4$, the existence of such a set $B\subset \mathbb{N}$ and $t\in \{0,1\}$ is assured. Furthermore, whenever $\underset{‾}{d}(A)>3\mathrm{/}4$ or $\bar{d}(A)>5\mathrm{/}6$, 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 $t\in \mathbb{N}$.