I am interested in a classification of sets $A \subseteq \mathbb{N}$ such that for all $k \in A$, $d( A+k \cap \mathbb{N} \setminus A) = 0$ where $d$ is the asymptotic density and $A+k = \{n \in \mathbb{N} : n-k \in A\}$. Sets with this condition are in a sense almost closed under addition. Some examples are sets of the form $C \setminus N$ where $C$ is closed under addition and $N$ has density $0$. Any set of density $0$ or $1$ also satisfies this property.
I hoped this condition was equivalent to $d(A \triangle C) = 0$ for some set $C$ closed under addition, but $A = 2 \mathbb{N} \cup \{1\}$ is a counterexample.
Is there some nice additive combinatorics classification of such sets?