Additive Complements for a Given Asymptotic Density

We investigate the existence of subsets A and B $${\mathbb {N}}:=\{0,1,2,\dots \}$$ , such that sumset $$A+B:=\{a+b:a\in A,b\in B\}$$ has prescribed asymptotic density. solve particular case in which is a given finite subset {N}}$$ also when $$B=A$$ ; later case, we generalize our result to $$kA:=\{x_1+\cdots +x_k: x_i\in A, i=1,\dots ,k\}$$ for an integer $$k\ge 2.$$