Additive Complements for a Given Asymptotic Density

Abstract

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 prescribed 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 \(k\ge 2.\)