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 $k\ge 2.$

13 pages