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

For a set A ⊂ N we characterize in terms of its density when there exists an infinite set B ⊂ N and t ∈ \0,1\ such that B+B ⊂ A-t, where B+B : =₁+b₂ b₁,b₂ ∈ B\. Specifically, when the lower density d(A) >1/2 or the upper density d̄(A)> 3/4, the existence of such a set B⊂ N and t∈ \0,1\ is assured. Furthermore, whenever d(A) > 3/4 or d̄(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 ⊂ N and number t ∈ N.