Additive Complements for a given Asymptotic Density

TL;DR

This paper explores the existence of subsets of natural numbers whose sumsets have specified asymptotic densities, providing solutions for cases where one summand is finite or when the sets are equal, and generalizing to multiple sums.

Contribution

It offers new results on constructing subsets with prescribed sumset densities, including finite and self-sum cases, extending to multiple sumsets.

Findings

Solved the case when B is finite.

Extended results to B=A and multiple sumsets.

Generalized to kA for k ≥ 2.

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 $k\geq2.$