A problem in additive number theory

For every real number $\alpha ,0 < \alpha < 1$, a sequence $A = \{ {a_1},{a_2}, \cdots \}$ is constructed for which the density of $A$ is $\alpha$ and $A$ has the following property: Given any $n$ distinct positive integers $\{ {b_1},{b_2}, \cdots ,{b_n}\}$ the sequence consisting of all numbers of the form ${a_i} + {b_j}$ has density $1 - {(1 - \alpha )^n}$.