Additive Complements for a given Asymptotic Density

Abstract: {The first version of this text was written and submitted to a journal on April, 12, 2018. This second version was submitted on April, 9, 2019.} We investigate the existence of subsets $A$ and $B$ of $\mathbb{N}:={0,1,2,\dots}$ such that the sumset $A+B:={a+b~;a\in A,b\in B}$ has given asymptotic density. We solve the particular case in which $B$ is a given finite subset of $\mathbb{N}$ and also the case when $B=A$ ; in the later case, we generalize our result to $kA:={x_1+\cdots+x_k: x_i\in A, i=1,\dots,k}$ for an integer $k\geq2.$