Inverse problem for upper asymptotic density

For a set $A$ of natural numbers, the structural properties are described when the upper asymptotic density of $2A+\{0,1\}$achieves the infimum of the upper asymptotic densities of all sets of the form $2B+\{0,1\}$, where the upper asymptotic density of $B$ is greater than or equal to the upper asymptotic density of $A$. As a corollary, we prove that if the upper asymptotic density of $A$ is less than $1$and the upper asymptotic density of $2A+\{0,1\}$ achieves the infimum, then the lower asymptotic density of $A$ must be $0$.