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: =\b₁+b₂ b₁, b₂ B\. Specifically, when the lower density d (A) >1/2 or the upper density d (A) > 3/4, the existence of such a set B N and t \0, 1\ is assured. Furthermore, whenever d (A) > 3/4 or 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 N and number t N.